Regularized mean curvature flow for invariant hypersurfaces in a

arXiv:1811.03441v1 [math.DG] 7 Nov 2018

Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory Naoyuki Koike

Abstract In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition and its image by the orbit map of the Hilbert Lie group action is included by the geodesic ball of some radius, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. As an application of this result to the gauge theory, we derive a result for some family of based gauge orbits in the space of connections of the principal bundle having a compact Lie group as the structure group over a compact Riemannian manifold.

1

Introduction

C. L. Terng ([Te1]) introduced the notion of a proper Fredholm submanifold in a (separable) Hilbert space as a submanifold of finite codimension satisfying certain conditions for the normal exponential map. Note that the shape operators of a proper Fredholm submanifold are compact operator. By using this fact, C. King and C. L. Terng ([KT]) defined the regularized mean curvature vector field of a proper Fredholm submanifold. Later, E. Heintze, C. Olmos and X. Liu (HLO]) defined another regularized mean curvature vector field of a proper Fredholm submanifold, which differs from the regularized mean curvature vector field defined in [KT]. They called the regularized mean curvature vector field defined in [KT] ζ-reguralized mean curvature vector field. The regularized mean curvature vector field in [HLO] is easier to handle than the the reguralized mean curvature vector field in [KT]. In almost

1

all relevant cases, these regularized mean curvature vector fields coincides. In this paper, we adopt the regularized mean curvature vector field defined in [HLO]. In this paper, we consider the regularized mean curvature flow in a (separable infinite dimensional) Hilbert space V . Let M be a Hilbert manifold and ft (0 ≤ t < T ) be a C ∞ -family of immersions of finite codimension of M into V . Assume that each ft is regularizable, where ”regularizability” means that ft is proper Fredholm and that, for each normal vector v of M , the regularized trace Trr (At )v of the shape operator (At )v of ft and the trace Tr (At )2v of (At )2v exist. See Section 3 about the definition of the regularized trace of the shape operator. Define a map F : M × [0, T ) → V by F (x, t) := ft (x) ((x, t) ∈ M × [0, T )). We call ft ’s (0 ≤ t < T ) the regularized mean curvature flow if the following evolution equation holds: (1.1)

∂F = △rt ft . ∂t

Here △rt ft is defined as the vector field along ft satisfying h△rt ft , vi := Trr h(∇t dft )(·, ·), vi♯ (∀ v ∈ V ), where ∇t is the Riemannian connection of the metric gt on M induced from the metric h , i of V by ft , h(∇t dft )(·, ·), vi♯ is the (1, 1)-tensor field on M defined by gt (h(∇t dft )(·, ·), vi♯ (X), Y ) = h(∇t dft )(X, Y ), vi (X, Y ∈ T M ) and Trr (·) is the regularized trace of (·). Denote by Ht the regularized mean curvature vector of ft (i.e., hHt , •i = Trr (At )• ). Note that △rt ft is equal to Ht . R. S. Hamilton ([Ha]) proved the existenceness and the uniqueness (in short time) of solutions of a weakly parabolic equation for sections of a finite dimensional vector bundle. The evolution equation (1.1) is regarded as the evolution equation for sections of the infinite dimensional trivial vector bundle M × V over M . Also, M is not compact. Hence we cannot apply the Hamilton’s result to this evolution equation (1.1). Also, we must impose certain kind of infinite dimensional invariantness for f because M is not compact. Thus, we cannot show the existenceness and the uniqueness (in short time) of solutions of (1.1) in general. However we ([Koi2]) showed the existenceness and the uniqueness (in short time) of solutions of (1.1) in the following special case. We consider a isometric almost free action of a Hilbert Lie group G on a Hilbert space V whose orbits are regularized minimal, that is, they are regularizable submanifold and their regularized mean curvature vectors vanish, where “almost free” means that the isotropy group of the action at each point is finite. Denote by N the orbit space V /G, which is an orbifold. Give N the Riemannian orbimetric gN such that the orbit map φ : V → N is a Riemannian orbisubmersion. Let M (⊂ V ) be a G-invariant submanifold in V . Assume that M := φ(M ) is compact. Denote by f the inclusion maps of M into V and f that of M into N . We ([Koi2]) showed that 2

the regularized mean curvature flow starting from M exists uniquely in short time. In this paper, we consider the case where M is an oriented hypersurface. Let ξ be a (global) unit normal vector field of M . Let K be the maximal sectional curvature of √ (N, gN ), which is nonnegative because V is flat. Set b := K. Let Σ be the singular set of (N, gN ) and {Σ1 , · · · , Σk } be the set of all connected components of Σ. Set BrT (x) := {v ∈ Tx M | kvk ≤ r}. For x ∈ N and r > 0, denote by Br (x) the geodesic ball of radius r centered at x. Here we note that, even if x ∈ Σ, the exponential map expx : Tx N → N is defined in the same manner as the Riemannian manifold-case and Br (x) is defined by Br (x) := expx (BrT (x)). We assume the following: (∗1 ) M is included by B πb (x0 ) for some x0 ∈ Σ and expx0 |B Tπ (x0 ) is injective. b

Furthermore, we assume the following: (∗2 )

2/n b2 (1 − α)−2/n ωn−1 · Volg¯ (M ) ≤ 1,

where α is a positive constant smaller than one and g denotes the induced metric on M and ωn denotes the volume of the unit ball in the Euclidean space Rn . Denote by f the inclusion map of M to V and f that of M into N . Let {ft }t∈[0,T ) be the regularized mean curvature flow starting from f . Denote by Ht the regularized mean curvature vector of ft and set Hts := −hHt , ξt i, where ξt is a unit normal vector field of ft such that ξ0 = ξ and t 7→ ξt is continuous. Set (Hts )min := minM Hts and (Hts )max := maxM Hts . Define a constant L by (♯)

L := sup

max

e1 )5u u∈V (X1 ,··· ,X5 )∈(H

e X Aφ )X X4 ), X5 i|, |hAφX1 ((∇ 2 3

e denotes the horizontal distribution of φ, (H e1 )u denotes the set of all unit where H φ horizontal vectors of φ at u and A denotes one of the O’Neill’s tensors defined in [O’N] (see Section 5 about the definition of Aφ ). Note that the restriction Aφ |H× e H e φ e e e of A to H × H is the tensor indicating the obstruction of the integrabilty of H. The main theorem in this paper is the following collapsing theorem. Theorem A. Assume that the initial G-invariant hypersurface f : M ֒→ V satisfies the above conditions (∗1 ), (∗2 ) and (H0s )2 (hH )0 > 2n2 L(gH )0 , where (gH )0 (resp. (hH )0 ) denotes the horizontal component of the induced metric (resp. the scalarvalued second fundamental form) of f . Then, for the regularized mean curvature flow {ft }t∈[0,T ) starting from f , the following statements (i) and (ii) hold: (i) T < ∞ and ft (M ) collapses to a G-orbit as t → T . (Hts )max = 1 holds. (ii) lim t→T (Hts )min 3

M

V

φ(M )

φ V /G

Figure 1 : Collapse to a G-orbit (Hts )max = 1” implies that ft (M ) converges to an infinitesimal t→T (Hts )min constant tube over some G-orbit as t → T (or equivalently, φ(ft (M )) converges to a round point(=an infinitesimal round sphere) as t → T ) (see Figure 2).

Remark 1.1. “ lim

φ(xmin (t)) φ(xmax (t)) xmax (t) : a point attaining (Hts )max xmin (t) : a point attaining (Hts )min

Figure 2 : Collapse of φ(ft (M )) to a round point Let π : P → M be a principal bundle having a compact semi-simple Lie group G as the structure group over a compact manifold. Give G the bi-invariant metric 0 induced from the Killing form B of the Lie algebra g of G. Let AH P be the (affine) 1 Hilbert space of all H 0 -connections of P and (GPH )x0 be the based gauge group of all H 1 -gauge transformations g’s of P at g|π−1 (x0 ) = e, where e is the identity element of G. Also, let H 0 ([0, 1], g) be the (separable) Hilbert space of all H 0 -paths in the Lie 1 1 algebra g of G and ΩH e (G) be the Hilbert Lie group of all H -loops g : [0, 1] → G’s 1 0 at e. The Hilbert Lie group ΩH e (G) acts on H ([0, 1], g) as follows: ′ (s ∈ [0, 1]) (g · u)(s) := AdG (g(s))(u(s)) − (Rg(s) )−1 ∗ (g (s)) 1 H (g ∈ Ωe (G), u ∈ H 0 ([0, 1], g)),

where AdG denotes the adjoint representation of G and Rg(s) is the right translation by g(s) and g′ denotes the weak derivative of g. Let c : [0, 1] → M be a H 1 -loop at 4

0

x0 and define a map holc : AH P → G by Pcω (u) = u · holc (ω)

(∀ u ∈ π −1 (x0 )),

where Pcω denotes the parallel translation along c with respect to ω. Note that 1 {holc (ω) | c ∈ ΩH x0 (M )} gives the holonomy group of the connection ω at x0 . Define a map ιc of the induced bundle c∗ P into P by ιc (t, u) = u ((t, u) ∈ c∗ P ). Let (π −1 (U ), ϕ) be a local trivialization of P . Assume that c([0, 1]) is included by U . The local trivialization ϕ gives the bundle isomorphism of the induced bundle π c : c∗ P → [0, 1] onto the trivial bundle π o : [0, 1] × G → [0, 1]. Denote by ϕc this bundle ∗ o isomorphism. (ιc ◦ ϕ−1 c ) ω gives a connection on the trivial bundle π : [0, 1] × G → 0 [0, 1], which is regaraded as an element of H ([0, 1], g) in a natural manner. Define 0 H0 −1 ∗ 0 c a map µcϕ : AH P → H ([0, 1], g) by µϕ (ω) := (ιc ◦ ϕc ) ω (ω ∈ AP ). As an application of Theorem A to the gauge theory, we obtain the following result. Theorem B. Let (π −1 (U ), ϕ) be a local trivialization of P and c : [0, 1] → U be an H 1 -loop at x0 . Let M be a strictly convex hypersurface in G satisfying the 0 conditions (∗1 ) and (∗2 ). Furthermore assume that M is included by holc (AH P ). Then the following statements (i) and (ii) hold. H1 (i) M := µcϕ (hol−1 c (M )) is a Ωe (G)-invariant regularizable hypersurface. 1 (ii) The regularized mean curvature flow starting from M collapses to a ΩH e (G)orbit in finite time. (iii) Let {Mt }t∈[0,T ) be the regularized mean curvature flow starting from M and set Bt := (µcϕ )−1 (Mt ). Then the flow {holc (Bt )}t∈[0,T ) collapses to a one-point set as t → T. The following question arises naturally. 0

Question. Let ιBt be the inclusion map of Bt into AH P . (i) Does there exist the regularized mean curvature vector Ht (in the sense of [MRT]) of ιBt ? ∂ιBt (ii) Does {Bt }t∈[0,T ) satisfy = Ht in the case where the regularized mean ∂t curvature vector Ht ’s exist?

2

Motivation from gauge theory

In this section, we state that the research in this paper leads to the research of subgauge orbits in the space of connections of the principal bundle having a compact 5

Lie group as the structure group over a compact Riemannian manifold. Y. Maeda, S. Rosenberg and P. Tondeur ([MRT]) investigated the submanifold geometry of the gauge orbits in the space of C ∞ -connections of a principal bundle having a compact Lie group as the structure group over a compact Riemannian manifold. They introduced the notion of the regularized mean curvature vector field of the gauge orbits. The regularized mean curvature vector field does not necessarily exist. Let π : P → M be a principal bundle with the structure group G over a compact Riemannian manifold (M, g), where G is a compact Lie group. Also, let A∞ P be the ∞ ∞ ∞ space of all C -connections of P and GP the group of all C -gauge transformations of P . The space A∞ echet space associated to the Fréchet space P is an affine Fr´ (M, Ad (P )), where Ad (P ) denotes the adjoint bundle P ×AdG g of P and Ω∞ G G 1 ∞ Ω1 (M, AdG (P )) denotes the space of all AdG (P )-valued 1-forms over M . Denote by Oω the gauge orbit GP∞ · ω through ω ∈ A∞ P . They showed the following facts: (i) If G is abelian, then Oω is totally geodesic (and hence the regularized mean curvature vector field of OP vanishes). (ii) If ω is flat, then the regularized mean curvature vector field of Oω exists. (iii) If dim M = 2 or odd, then the regularized mean curvature vector field of Oω exists for any ω ∈ A∞ P . (iv) In the case of dim M = 4, the regularized mean curvature vector field of Oω exists if and only if ω is a Yang-Mills connection. These facts state in Lemma 5.1 and Thereom 5.10 of [MRT]. Also the following facts hold: set.

∞ (v) If dim M = 1 (i.e., M = S 1 ), then the muduli space A∞ P /GP is a one-point

∞ (vi) If dim M ≥ 2, then the muduli space A∞ P /GP is an infinite dimensional space with singularities.

The Riemannian Fréchet submanifold theory is difficult to treat in comparison with the Riemannian Hilbert submanifold theory. So we had better con0 0 sider the above research in the Hilbert setting. Let AH P be the space of all H 1 connections of P and GPH the group of all H 1 -gauge transformations of P . The 0 space AH P is an affine Hilbert space associated to the (separable) Hilbert space 0 2 H0 ΩH 1 (M, AdG (P )) equipped with the natural L -inner product. By identifying AP 0 H0 with ΩH 1 (M, AdG (P )), we regard AP as a (separable) Hilbert space. Denote by 1 0 Oω the gauge orbit GPH ·ω through ω ∈ AH P . The gauge orbits Oω ’s are Riemannian 0 Hilbert submanifolds in AH P . The above facts (i)-(vi) hold in this Hilbert setting. In this paper, we investigate invariant hypersurfaces in a (separable) Hilbert space V equipped with almost free Hilbert Lie group action G y V satisfying the following 6

two conditions: (i) Each G-orbit has the vanishing regularized mean curvature vector field, (ii) The orbit space V /G is a finite dimensional compact orbifold, where “invariant” means that the hypersurface is invariant with respect to the G0 action. Hence we want to find the pair (B; G1 ) of a closed subspace B of AH P and a 1 closed subgroup G1 of GPH satisfying (I) Fibres of the orbit map φ : B → B/G1 have the vanishing regularized mean curvature vector field. (II) The action of G1 on B is almost free and the moduli space B/G1 is a finite dimensional compact orbifold. Here we give examples of the pair (B, G1 ) satisfying the conditions (I) and (II). We consider the trivial bundle P0 := [0, 1]×G over the closed interval [0, 1]. Assume that 0 H0 G is a compact semi-simple Lie group. Then AH P0 (≈ Ω1 (M, AdG (P0 ))) is identified 1 with the space H 0 ([0, 1], g) of all H 0 -paths in the Lie algebra g of G and GPH0 is identified with the Hilbert Lie group H 1 ([0, 1], G) of all H 1 -paths in G. For closed subgroups H and K of G, we define a closed subgroup P (G, H × K) of H 1 ([0, 1], G) by P (G, H × K) := {g ∈ H 1 ([0, 1], G) | (g(0), g(1)) ∈ H × K}. 1

In particular, denote by ΩH e (G) the loop group P (G, {e} × {e}) at the idenitity element e of G. Let Γ be a finite group of G and K a closed subgroup of G such that (G, K) is a reductive pair, that is, there exists a subspace p of g satisfying g = k ⊕ p and [k, p] ⊂ p, where g and k denote the Lie algebras of G and K, respectively. 0 Let B := AH P and G1 := P (G, Γ × K). Then the pair (B, G1 ) satisfies the above conditions (I) and (II), and the moduli space B/G1 is diffeomorphic to the orbifold Γ \ G / K, where we note that, if Γ acts on G/K freely, then B/G1 = Γ \ G / K is a C ∞ -manifold. We shall state that the research of P (G, H × K)-invariant submanifolds in 0 0 H ([0, 1], g) leads to that of certain kind of subgauge orbits in AH P . Let π : P → M be a G-bundle over M . Denote by H 1 ([0, 1], M ) the Hilbert manifold of all H 1 -paths 1 in M . First we shall consider the case where P (G, H × K) is the loop group ΩH e (G). For x ∈ M , we set 1 Gx := {g ∈ GPH | g|π−1 (x) = id}, which is called the based gauge group at x. It is clear that Gx is a normal subgroup 0 of G. It is known that, if dim M ≥ 2, then AH P /Gx is an infinite dimensional 1 Hilbert manifold (without singularity). For an open set U of M , we set ΩH x (U ) := {c ∈ H 1 ([0, 1], U ) | c(0) = c(1) = x} (the space of all H 1 -loops at x in U ) and 7

1

1 1 ΩH e (G) := {g ∈ H ([0, 1], G) | g(0) = g(1) = e} (the H -loop group of G at e). Take 1 a local trivialization (π −1 (U ), ϕ) of π : P → M about x ∈ U and c ∈ ΩH x (U ). Let c, where ιc , ϕc and µcϕ be as stated in Introduction. For g ∈ Gx , we set gϕc := g◦ϕ−1 ◦b 1 c is the H -curve in U × G defined by b b c(t) := (c(t), e) (t ∈ [0, 1]). It is clear that gϕc 1 c c H1 c is an element of ΩH e (G). Define a map λϕ : Gx → Ωe (G) by λϕ (g) := gϕ (g ∈ Gx ). It is clear that 0

µcϕ (g · ω) = λcϕ (g) · µcϕ (ω) 0

(g ∈ Gx , ω ∈ AH P ). 1

H 0 Therefore µcϕ maps Gx -orbits in AH P to Ωe (G)-orbits in H ([0, 1], g) and hence it 0 0 H1 induces the map µcϕ between the orbits spaces AH P /Gx and H ([0, 1], g)/Ωe (G). 1 0 H0 Thus the research of ΩH e (G)-orbits in H ([0, 1], g) leads to that of Gx -orbits in AP through µcϕ ’s.

λcϕ

Gx

1

ΩH e (G) µcϕ

0 AH P

H 0 ([0, 1], g)

holc µcϕ

0

AH P /Gx

φ 1

H 0 ([0, 1], g)/ΩH e (G) = G

1

Figure 3 : ΩH e (G)-orbits and Gx -orbits The orbit map φ : H 0 ([0, 1], g) → H 0 ([0, 1], g)/Ωe (G) = G is called the parallel transport map for G. See [Te1], [Te2], [PiTh], [PaTe], [KT], [HLO], [Koi1] etc about the research of the parallel trandsport map. Next we shall consider the case where P (G, H × K) is general. For u1 , u2 ∈ P , we set 1 Gu1 ,H;u2 ,K := {g ∈ GPH | g(u1 ) ∈ H, g(u2 ) ∈ K}.

Also, for an open set U of M and x1 , x2 ∈ U , we set

P (U, {x1 } × {x2 }) := {c ∈ H 1 ([0, 1], U ) | c(0) = x1 , c(1) = x2 }.

Assume that the restricted bundle P |U is trivial. Then we can take a local trivialization (π −1 (U ), ϕ) of P such that ϕ(ui ) = (xi , e) (i = 1, 2). For c ∈ P (U, {x1 } × {x2 }) and g ∈ Gu1 ,H;u2,K , we set gϕc := g ◦ ϕ−1 ◦ b c. It is clear that gϕc is an element of c P (G, H × K). Hence a map λϕ : Gu1 ,H;u2 ,K → P (G, H × K) is defined by λcϕ (g) := gϕc

(g ∈ Gu1 ,H;u2 ,K ). 8

It is clear that 0

µcϕ (g · ω) = λcϕ (g) · µcϕ (ω)

(g ∈ Gu1 ,H;u2,K , ω ∈ AH P ). 0

0 Therefore µcϕ maps Gu1 ,H;u2 ,K -orbits in AH P to P (G, H × K)-orbits in H ([0, 1], g) 0 and hence it induces the map µcϕ between the orbits spaces AH P /Gu1 ,H;u2 ,K and H 0 ([0, 1], g)/P (G, H × K). Thus the research of P (G, H × K)-orbits in H 0 ([0, 1], g) 0 c leads to that of Gu1 ,H;u2 ,K -orbits in AH P through µϕ ’s. 0 At the end of this section, we shall give a representation of the space AH P into 1 the infinite product of H 0 ([0, 1], g)’s. For an open set U of M , set ΩH (U ) := {c ∈ H 1 ([0, 1], U ) | c(0) = c(1)}. Fix a finite family {(π −1 (Ui ), ϕi )}ki=1 of the local trivializations of P satisfying

(i) {Ui }ki=1 is an open cevovering of M , (ii) {Ui }i∈B is not an open covering of M for any proper subset B of {1, · · · , k}. Set

k

1

S := ∐ {(c, i) | c ∈ ΩH (Ui )}. i=1

By using µcϕi ’s, we define a map ρ by 0

♯S 0 0 ρ := (µcϕi )(c,i)∈S : AH P → Π H ([0, 1], g)(= Map(S, H ([0, 1], g))),

where ♯S denotes the cardinality of S. It is clear that ρ is injective. This injection 0 ρ is interpreted as a representation of AH P into the (infinite) product space of the H 0 -path spaces in g. We shall compare this representation ρ with the representation given by S. Kobayashi ([Kob]). First we shall recall the Kobayashi’s representation. Define an equivalence relation ∼el in H 1 ([0, 1], M ) by −1 c ∼el cˆ : ⇔ cˆ = c · c1 · c−1 for some c1 , · · · , ck ∈ H 1 ([0, 1], M ), 1 · · · · · ck · ck def

where c−1 denotes the inverse path of ci and · denotes the product of paths. The i 1 H quotient Ωx (M )/ ∼el is a group. The Kobayashi’s representation is the monomor0 H1 phism ρ : AH P /Gx → Hom(Ωx (M )/ ∼el , G) defined by ρ(Gx · ω) : [c]el 7→ holc (ω), where holc is as stated in Introduction. Between ρ and ρ, the following relation holds: 0 φ(ρ(ω)(c, i)) = ρ(Gc(0) · ω)([c]el ) (ω ∈ AH P , (c, i) ∈ S). 9

3

The regularized mean curvature flow

Let f be an immersion of an (infinite dimensional) Hilbert manifold M into a Hilbert space V and A the shape tensor of f . If codim M < ∞, if the differential of the normal exponential map exp⊥ of f at each point of M is a Fredholm operator and if the restriction exp⊥ to the unit normal ball bundle of f is proper, then M is called a proper Fredholm submanifold. In this paper, we then call f a proper Fredholm immersion. Then the shape operator Av is a compact operator for each normal vector v of M . Furthermore, if, for each normal vector v of M , the regularized trace Trr Av and Tr A2v exist, then M is called regularizable submanifold, where Trr Av ∞ P − − − + + is defined by Trr Av := (µ+ i + µi ) (µ1 ≤ µ2 ≤ · · · ≤ 0 ≤ · · · ≤ µ2 ≤ µ1 : i=1

the spectrum of Av ). In this paper, we then call f regularizable immersion. If f is a regulalizable immersion, then the regularized mean curvature vector H of f is defined as the normal vector field satisfying hH, vi = Trr Av (∀ v ∈ T ⊥ M ), where h , i is the inner product of V and T ⊥ M is the normal bundle of f . If H = 0, then f is said to be minimal. In particular, we consider the case where f is of codimension one and it admits a global unit normal vector field. Fix a global unit normal vector field ξ. Then we call −hH, ξi the regularized mean curvature of f and denote it by H s . Also, we call −Aξ the shape operator and denote it by the same symbol A. Remark 3.1. In the research of the mean curvature flow starting from strictly convex hypersurfaces, it is general to take the outward unit normal vector field as the unit normal vector field ξ and −Aξ as the shape operator and −hH, ξi as the mean curvature. Hence we take the shape operator A and the regularized mean curvature H s as above. Let ft (0 ≤ t < T ) be a C ∞ -family of regularizable immersions of M into V . Denote by Ht the regularized mean curvature vector of ft . Define a map F : M × [0, T ) → V by F (x, t) := ft (x) ((x, t) ∈ M × [0, T )). If ∂F ∂t = Ht holds, then we call ft (0 ≤ t < T ) the regularized mean curvature flow.

4

The mean curvature flow in Riemannian orbifolds

In this section, we shall recall the notion of the mean curvaure flow starting from a suborbifold in a Riemannian orbifold introduced in [Koi2]. First we recall the notions of a Riemannian orbifold and a suborbifold following to [AK,BB,GKP,Sa,Sh,Th]. e /Γ) a triple satisfying the Let M be a paracompact Hausdorff space and (U, ϕ, U following conditions: 10

(i) U is an open set of M , b is an open set of Rn and Γ is a finite subgroup of the C k -diffeomorphism (ii) U b ) of U b, group Diff k (U b /Γ. (iii) ϕ is a homeomorphism of U onto U

b /Γ) is called an n-dimensional orbifold chart. Let O := Such a triple (U, ϕ, U b {(Uλ , ϕλ , U /Γλ ) | λ ∈ Λ} be a family of n-dimensional orbifold charts of M satisfying the following conditions:

(O1) {Uλ | λ ∈ Λ} is an open covering of M , (O2) For λ, µ ∈ Λ with Uλ ∩ Uµ 6= ∅, there exists an n-dimensional orbifold chart c /Γ′ ) such that C k -embeddings ρλ : W c ֒→ U bλ and ρµ : W c ֒→ U bµ (W, ψ, W −1 −1 ◦ π ′ , where satisfying ϕλ ◦ πΓλ ◦ ρλ = ψ −1 ◦ πΓ′ and ϕ−1 Γ µ ◦ πΓµ ◦ ρµ = ψ ′ πΓλ , πΓµ and πΓ′ are the orbit maps of Γλ , Γµ and Γ , respectively.

Such a family O is called an n-dimensional C k -orbifold atlas of M and the pair bλ /Γλ ) be an n-dimensional (M, O) is called an n-dimensional C k -orbifold. Let (Uλ , ϕλ , U orbifold chart around x ∈ M . Then the group (Γλ )xb := {b ∈ Γλ | b(b x) = x b} is unique bλ with (ϕ−1 ◦πΓ )(b x ) = x. Denote for x up to the conjugation, where x b is a point of U λ λ by (Γλ )x the conjugate class of this group (Γλ )xb , This conjugate class is called the local group at x. If the local group at x is not trivial, then x is called a singular point of (M, O). Denote by Sing(M, O) (or Sing(M )) the set of all singular points of (M, O). This set Sing(M, O) is called the singular set of (M, O). Let x ∈ M and bλ /Γλ ) an orbifold chart around x. Take x bλ with (ϕ−1 ◦ πΓ )(b xλ ) = x. (Uλ , ϕλ , U bλ ∈ U λ λ b The group (Γλ )xbλ acts on Txbλ Uλ naturally. Denote by Ox the subfamily of OM conbλ )/(Γλ )xb an (Txbλ U sisting of all orbifold charts around x. Give Tx := ∐ λ bλ /Γλ )∈Ox (Uλ ,ϕλ ,U

bλ /Γλ ) and (Uλ , ϕλ , U bλ /Γλ ) be equivalence relation ∼ as follows. Let (Uλ1 , ϕλ1 , U 1 1 2 2 2 2 members of OM,x . Let η be the diffeomorphism of a sufficiently small neighborhood bλ into U bλ satisfying ϕ−1 ◦ πΓ ◦ η = ϕ−1 ◦ πΓ . Define an equivalence of x bλ1 in U 1 2 λ2 λ1 λ2 λ1 relatiton ∼ in Tx as the relation generated by [v1 ] ∼ [v2 ] ⇐⇒ [v2 ] = [η∗ (v1 )] def b /(Γλ )xb ), bλ /(Γλ )xb , [v2 ] ∈ Txb U ([v1 ] ∈ Txbλ1 U 2 1 1 λ2 λ2 λ2 λ1

bλ . We call the quotient where [vi ] (i = 1, 2) is the (Γλi )xbλi -orbits through vi ∈ Txbλi U i space Tx / ∼ the orbitangent space of M at x and denote it by Tx M . If (M, O) is of class C k (k ≥ 1), then T M := ∐ Tx M is a C k−1 -orbifold in a natural manner. x∈M

We call T M the orbitangent bundle of M . The group (Γλ )xbλ acts on the (r, s)(r,s) b (r,s) b b tensor space Txbλ U λ of Tx bλ Uλ naturally. Denote by (Tx bλ Uλ )(Γλ )xb the space of all λ

11

(r,s) b (Γλ )xbλ -invariant elements of Txbλ U λ and set

Tx(r,s) :=

∐

bλ /Γλ )∈Ox (Uλ ,ϕλ ,U

(r,s) b (Txbλ U λ )(Γλ )xb . λ

bλ /Γλ ) ∈ Ox (i = 1, 2). Let η be C r -diffeomorphism of a Take any (Uλi , ϕλi , U i i bλ onto an open subset of U bλ satissufficiently small neighborhood of x bλ1 ∈ U 1 2 (r,s) b −1 −1 fying ϕλ2 ◦ πΓλ2 ◦ η = ϕλ1 ◦ πΓλ1 . Among elements of (Txbλ Uλ1 )(Γλ1 )xb and 1

(r,s) b (Txbλ U λ2 )(Γλ 2

) bλ 2 x 2

λ1

, we consider the following relation∼λ1 ,λ2 : S1 ∼λ1 ,λ2 S2 ⇐⇒ S2 = η ∗ (S1 )

(S1 ∈

def (r,s) b (Tpbλ Uλ1 )(Γλ1 )pb , S2 λ1 1

(r,s) b ∈ (Tpbλ U λ2 )(Γλ2 )pb ). 2

λ2

(r,s)

generetaed by ∼λ1 ,λ2 . We call the Denote by ∼ the equivalence relation in Tx (r,s) quotient set Tx / ∼ the (r, s)-tensor orbitensor space of M at x and denote by (r,s) Tx M . If (M, O) is of class C k (k ≥ 1), then T (r,s) M := ∐ Tx(r,s) M is a C k−1 x∈M

orbifold in a natural manner. We call T (r,s) M the (r, s)-orbitensor bundle of M . Let (M, OM ) and (N, ON ) be orbifolds, and f a map from M to N . If, for each bλ /Γλ ) of (M, OM ) around x x ∈ M and each pair of an orbifold chart (Uλ , ϕλ , U ′ and an orbifold chart (Vµ , ψµ , Vbµ /Γµ ) of (N, ON ) around f (x) (f (Uλ ) ⊂ Vµ ), there bλ → Vbµ with f ◦ ϕ−1 ◦ πΓ = ψµ−1 ◦ πΓ′ ◦ fbλ,µ , then f is exists a C k -map fbλ,µ : U λ λ µ k k b called a C -orbimap (or simply a C -map). Also fλ,µ is called a local lift of f with bλ /Γλ ) and (Vµ , ψµ , Vbµ /Γ′µ ). Furthermore, if each local lift fbλ,µ respect to (Uλ , ϕλ , U

is an immersion, then f is called a C k -orbiimmersion (or simply a C k -immersion) and (M, OM ) is called a C k -(immersed) suborbifold in (N, ON , g). Similarly, if each local lift fbλ,µ is a submersion, then f is called a C k -orbisubmersion. In the sequel, we assume that r = ∞. Denote by prT M and prT (r,s) M the natural projections of T M and T (r,s) M onto M , respectively. These are C ∞ -orbimaps. We call a C k -orbimap X : M → T M with prT M ◦ X = id a C k -orbitangent vector field on (M, OM ) and a C k -orbimap S : M → T (r,s) M with prT (r,s) M ◦ S = id a (r, s)bλ /Γλ ), an orbitensor field of class C k on (M, OM ). For an orbifold chart (Uλ , ϕλ , U −1 orbifolds chart of T (r,s) M having prT (r,s) M (Uλ ) as the domain is uniquely defined. bλ /Γλ ) Denote by Sbλ the local lift of (r, s)-orbitensor field S with respect to (Uλ , ϕλ , U −1 and the orbifold chart having prT (r,s) M (Uλ ) as the domain. If a (r, s)-orbitensor field g of class C k on (M, OM ) is positive definite and symmetric, then we call g a C k Riemannian orbimetric and (M, OM , g) a C k -Riemannian orbifold. 12

Let f be a C ∞ -orbiimmersion of an C ∞ -orbifold (M, OM ) into C ∞ -Riemannian bλ /Γλ ) of M around x and an orbifold (N, ON , g). Take an orbifold chart (Uλ , ϕλ , U ′ orbifold chart (Vµ , ψµ , Vbµ /Γµ ) of N around f (x) with f (Uλ ) ⊂ Vµ . Let fbλ,µ be the local lift of f with respect to these orbifold charts and gbµ that of g to Vbµ . Denote bλ )µ the orthogonal complement of (fbλ,µ )∗ (Txb U b ) in (T d Vbµ , (b gµ )fd ). by (Txb⊥λ U λ λ (x) f (x) µ

µ

Give

Tx⊥ :=

∐

∐

bλ /Γλ )∈OM,x (Vµ ,ψµ ,Vbµ /Γµ )∈ON,f (x) (Uλ ,ϕλ ,U

bλ )µ /(Γ′µ )⊥λ (Txb⊥λ U d

f (x)µ

λ denotes the subgroup of (Γ′µ )fd an equivalence relation ∼ as follows, where (Γ′µ )⊥d (x)

f (x)µ

consisting of elements preserving

bλ )µ (Txb⊥λ U

µ

bλ /Γλ ) (i = invariantly. Let (Uλi , ϕλi , U i i

1, 2) be members of OM,x and (Vµi , ψµi , Vbµi /Γ′µi ) (i = 1, 2) members of ON,f (x) with f (Uλi ) ⊂ Vµi . Let ηµ1 ,µ2 be the diffeomorphism of a sufficiently small neighborhood of fd (x)µ1 in Vbµ1 into Vbµ2 satisfying ψµ−1 ◦ πΓ′µ ◦ ηµ1 ,µ2 = ψµ−1 ◦ πΓ′µ . Define an 2 1 2 1 ⊥ equivalence relatiton ∼ in Tx as the relation generated by [ξ1 ] ∼ [ξ2 ] ⇐⇒ [ξ2 ] = [(ηµ1 ,µ2 )∗ (ξ1 )] def

bλ )µ /(Γ′ )⊥λ1 ([ξ1 ] ∈ (Txb⊥λ U µ1 d 1 1

f (x)µ

1

1

⊥λ

bλ )µ /(Γ′ ) 2 , [ξ2 ] ∈ (Txbλ2 U 2 µ2 d 2

f (x)µ

),

2

⊥λ bλ )µ ). We call where [ξi ] (i = 1, 2) denotes the (Γ′µi ) di -orbit through ξi ∈ (Txb⊥λ U i i f (x)µ

i

i

the quotient space Tx⊥ / ∼ the orbinormal space of M at x and denote it by Tx⊥ M . If f is of class C ∞ , then T ⊥ M := ∐ Tx⊥ M is a C ∞ -orbifold in a natural manner. x∈M

We call T ⊥ M the orbinormal bundle of M . Denote by prT ⊥ M the natural projection of T ⊥ M onto M . This is C ∞ -orbisubmersion. We call a C k -orbimap ξ : M → T ⊥ M with prT ⊥ M ◦ ξ = id a C k -orbinormal vector field of (M, OM ) in (N, ON , g). bλ /Γλ ) of M around x and an orbifold chart Take an orbifold chart (Uλ , ϕλ , U bλ )(r,s) the (r, s)(Vµ , ψµ , Vbµ /Γ′µ ) of N around f (x) with f (Uλ ) ⊂ Vµ . Denote by (Txb⊥λ U µ (r,s) ⊥λ ⊥ ′ ⊥ b b acts on (T Uλ )µ naturally. Denote tensor space of (T Uλ )µ . The group (Γµ ) x bλ

by

bλ )(r,s) ((Txb⊥λ U µ )(Γ′ )⊥λ µ

the space of all

fd (x)µ

x bλ fd (x)µ λ -invariant (Γ′µ )⊥d f (x)µ

and set

(Tx⊥ )(r,s) :=

∐

∐

bλ /Γλ )∈OM,x (Vµ ,ψµ ,Vbµ /Γµ )∈ON,f (x) (Uλ ,ϕλ ,U

(r,s)

bλ )µ elements of (Txbλ U

bλ )(r,s) ) . ((Txb⊥λ U ⊥ µ (Γ′ ) λ µ

fd (x)µ

bλ /Γλ ) and (Vµ , ψµ , Vbµ /Γ′ ) (i = 1, 2) and ηµ ,µ as above. Among Take (Uλi , ϕλi , U µi 1 2 i i i i i (r,s) (r,s) ⊥ ⊥ b b elments of ((T Uλ )µ1 ) and ((T Uλ )µ2 ) , we consider the ⊥λ ⊥λ x bλ1

1

x bλ2

(Γ′µ1 ) d1

f (x)µ 1

13

2

(Γ′µ2 ) d2

f (x)µ 2

following relation ∼λ1 ,λ2 ;µ1 ,µ2 : S1 ∼λ1 ,λ2 ;µ1 ,µ2 S2 ⇐⇒ S2 = (ηµ1 ,µ2 )∗ (S1 ) def bλ )(r,s) bλ )(r,s) (S1 ∈ ((Txb⊥λ U ) , S2 ∈ ((Txb⊥λ U ⊥λ 1 µ1 2 µ2 ) 1 ′ 1

(Γµ1 ) d

2

f (x)µ 1

⊥λ

(Γ′µ2 ) d2

).

f (x)µ 2

Let ∼ be the equivalence relation in (Tx⊥ )(r,s) generetaed by ∼λ1 ,λ2 . We denote the quotient set (Tx⊥ )(r,s) / ∼ by (Tx⊥ M )(r,s) . If f is of class C ∞ , then (T ⊥ M )(r,s) := ∐ (Tx⊥ M )(r,s) is a C ∞ -orbifold in a natural manner. We call (T ⊥ M )(r,s) the x∈M

(r, s)-orbitensor bundle of T ⊥ M . Denote by pr(T ⊥ M )(r,s) the natural projection of (T ⊥ M )(r,s) onto M . This is C ∞ -orbisubmersion. Similarly, we can define the or′ ′ ′ ′ bitensor product bundle T (r,s) M ⊗ (T ⊥ M )(s ,t ) of T (r,s) M and (T ⊥ M )(s ,t ) . Denote ′ ′ by prT (r,s) M ⊗(T ⊥ M )(s′ ,t′ ) the natural projection of T (r,s)M ⊗ (T ⊥ M )(s ,t ) onto M . ′

′

This is a C ∞ -orbisubmersion. We call a C k -orbimap S : M → T (r,s) M ⊗(T ⊥ M )(s ,t ) ′ ′ with prT (r,s) M ⊗(T ⊥ M )(s′ ,t′ ) ◦ S = id a C k -section of T (r,s) M ⊗ (T ⊥ M )(s ,t ) . Let g, h, A, H and ξ be the induced metric, the second fundamental form, the shape tensor, the mean curvature and a unit normal vector field of the immersion f |M \Sing(M ) : M \ Sing(M ) ֒→ N \ Sing(N ), respectively. It is easy to show that g, h, A and H extend a (0, 2)-orbitensor field of class C ∞ on (M, OM ), a C k -section of T (0,2) M ⊗ T ⊥ M , a C k -section of T (1,1) M ⊗ (T ⊥ M )(0,1) and a C ∞ -orbinormal vector field on (M, OM ). We denote these extensions by the same symbols. We call these extensions g, h, A and H the induced orbimetric, the second fundamental orbiform, the shape orbitensor and the mean curvature orbifunction of f . Here we note that ξ does not necessarily extend a C ∞ -orbinormal vector field on (M, O). Now we shall define the notion of the mean curvature flow starting from a C ∞ suborbifold in a C ∞ -Riemannian orbifold. Let ft (0 ≤ t < T ) be a C ∞ -family of C ∞ -orbiimmersions of a C ∞ -orbifold (M, OM ) into a C ∞ -Riemannian orbifold (N, ON , g). Assume that, for each (x0 , t0 ) ∈ M × [0, T ) and each pair of an orbifold bλ /Γλ ) of (M, OM ) around x0 and an orbifold chart (Vµ , ϕµ , Vbµ /Γ′µ ) chart (Uλ , ϕλ , U of (N, ON ) around ft0 (x0 ) such that ft (Uλ ) ⊂ Vµ for any t ∈ [t0 , t0 + ε) (ε : a bλ → Vbµ of ft sufficiently small positive number), there exists local lifts (fbt )λ,µ : U (t ∈ [t0 , t0 + ε)) such that they give the mean curvature flow in (Vbµ , gbµ ), where gbµ is the local lift of g to Vbµ . Then we call ft (0 ≤ t < T ) the mean curvature flow in (N, ON , g). In [Koi2], the following fact was shown for this flow. Theorem 4.1([Koi2]). For any C ∞ -orbiimmersion f of a compact C ∞ -orbifold into a C ∞ -Riemannian orbifold, the mean curvature flow starting from f exists uniquely in short time. 14

5

Evolution equations

Let G y V be an isometric almost free action with minimal regularizable orbit of a Hilbert Lie group G on a Hilbert space V equipped with an inner product h , i. The orbit space V /G is a (finite dimensional) C ∞ -orbifold. Let φ : V → V /G be the orbit map and set N := V /G. Here we give an example of such an isometric almost free action of a Hilbert Lie group. Example. Let G be a compact semi-simple Lie group, K a closed subgroup of G and Γ a discrete subgroup of G. Denote by g and k the Lie algebras of G and K, respectively. Assume that a reductive decomposition g = k + p exists. Let B be the Killing form of g. Give G the bi-invariant metric induced from B. Let H 0 ([0, 1], g) be the Hilbert space of all paths in the Lie algebra g of G which are L2 -integrable with respect to B. Also, let H 1 ([0, 1], G) the Hilbert Lie group of all paths in G which are of class H 1 with respect to g. This group H 1 ([0, 1], G) acts on H 0 ([0, 1], g) isometrically and transitively as a gauge action: ′ (g · u)(s) = AdG (g(s))(u(s)) − (Rg(s) )−1 ∗ (g (s)) (s ∈ [0, 1]) 1 (g ∈ H ([0, 1], G), u ∈ H 0 ([0, 1], g)).

Set P (G, Γ×K) := {a ∈ H 1 ([0, 1], G) | (a(0), a(1)) ∈ Γ×K}. The group P (G, Γ×K) acts on H 0 ([0, 1], g) almost freely and isometrically, and the orbit space of this action is diffeomorphic to the orbifold Γ \ G / K. Furthermore, each orbit of this action is regularizable and minimal. Give N the Riemannian orbimetric such that φ is a Riemannian orbisubmersion. Let f : M ֒→ V be a G-invariant submanifold immersion such that (φ ◦ f )(M ) is compact. For this immersion f , we can take an orbiimmesion f of a compact orbifold M into N and an orbisubmersion φM : M → M with φ ◦ f = f ◦ φM . Let ft (0 ≤ t < T ) be the regularized mean curvature flow starting from f and f t (0 ≤ t < T ) the mean curvature flow starting from f . The existence and the uniqueness of these flows is assured by Theorem 4.1 and Proposition 4.1 in [Koi2]. Define a map F : M × [0, T ) → V by F (u, t) := ft (u) ((u, t) ∈ M × [0, T )) and a map F : M × [0, T ) → N by F (x, t) := f t (x) ((x, t) ∈ M × [0, T )). Denote by Ht the regularized mean curvature vector of ft and H t the mean curvature vector of f t . Since φ has minimal regularizable fibres, Ht is the horizontal lift of H t , we can show that φ◦ft = f t ◦φM holds for all t ∈ [0, T ). In the sequel, we consider the case where e (resp. V) e the horizontal (resp. the codimension of M is equal to one. Denote by H vertical) distribution of φ. Denote by prHe (resp. prVe ) the orthogonal projection of e (resp. V). e For simplicity, for X ∈ T V , we denote pr e (X) (resp. pr e (X)) T V onto H V H 15

ef (u) by XHe (resp. XVe ). Define a distribution Ht on M by ft∗ ((Ht )u ) = ft∗ (Tu M )∩ H t ef (u) (u ∈ M ). Note that Vt is (u ∈ M ) and a distribution Vt on M by ft∗ ((Vt )u ) = V t independent of the choice of t ∈ [0, T ). Fix a unit normal vector field ξt of ft . Denote by gt , ht , At , Ht and Hts the induced metric, the second fundamental form (for −ξt ), the shape operator (for −ξt ) and the regularized mean curvature vector and the regularized mean curvature (for −ξt ), respectively. The group G acts on M through e is a connection of this orbibundle, it ft . Since φ : V → V /G is a G-orbibundle and H follows from Proposition 5.1 that this action G y M is independent of the choice of t ∈ [0, T ). It is clear that quantities gt , ht , At , Ht and Hts are G-invariant. Also, let ∇t be the Riemannian connection of gt . Let πM be the projection of M × [0, T ) onto ∗ E the induced bundle of E by π . M . For a vector bundle E over M , denote by πM M ∗ (T (0,2) M ) Also denote by Γ(E) the space of all sections of E. Define a section g of πM by g(u, t) = (gt )u ((u, t) ∈ M × [0, T )), where T (0,2) M is the (0, 2)-tensor bundle of ∗ (T (0,2) M ), a section A of π ∗ (T (1,1) M ), M . Similarly, we define a section h of πM M ∗ a section H of F T V and a section ξ of F ∗ T V . We regard H and ξ as V -valued functions over M × [0, T ) under the identification of Tu V ’s (u ∈ V ) and V . Define a ∗ T M by H subbundle H (resp. V) of πM (u,t) := (Ht )u (resp. V(u,t) := (Vt )u ). Denote ∗ (T M ) onto H (resp. V). For by prH (resp. prV ) the orthogonal projection of πM ∗ simplicity, for X ∈ πM (T M ), we denote prH (X) (resp.prV (X)) by XH (resp. XV ). dB(u,t) ∂B ∂B ∗ (T (r,s) M ), we define by , where := For a section B of πM ∂t ∂t (u,t) dt the right-hand side of this relation is the derivative of the vector-valued function (r,s) ∗ (T (r,s) M ) by t 7→ B(u,t) (∈ Tu M ). Also, we define a section BH of πM BH = (prH ⊗ · · · ⊗ prH ) ◦B ◦ (prH ⊗ · · · ⊗ prH ) . (r−times)

(s−times)

The restriction of BH to H × · · · × H (s-times) is regarded as a section of the (r, s)-tensor bundle H(r,s) of H. This restriction also is denoted by the same symbol BH . Let DM (resp. D[0,T ) ) be the subbundle of T (M × [0, T )) defined by (DM )(u,t) := T(u,t) (M × {t}) (resp. (D[0,T ) )(u,t) := T(x,t) ({u} × [0, T )) for each L (u, t) ∈ M × [0, T ). Denote by v(u,t) the horozontal lift of v ∈ Tu M to (u, t) with L L ) = v). respect to πM (i.e., v(u,t) is the element of (DM )(u,t) with (πM )∗(u,t) (v(u,t) L Under the identification of ((u, t), v)(= v) ∈ (π ∗ T M )(u,t) with v(u,t) ∈ (DM )(u,t) , we ∗ identify πM T M with DM . For a tangent vector field X on M (or an open set U ∗ T M )(= Γ(D )) (or Γ((π ∗ T M )| )(= Γ((D )| ))) by of M ), we define X ∈ Γ(πM M U M U M L e X (u,t) := ((u, t), Xu )(= (Xu )(u,t) ) ((u, t) ∈ M ×[0, T )). Denote by ∇ the Riemannian ∗ T M defined by connection of V . Let ∇ be the connection of πM (∇X Y )(u,t) := ∇tX(u,t) Y(·,t) and (∇ ∂ Y )(u,t) := ∂t

16

dY(u,·) dt

∗ T M ), where X for X ∈ Γ(DM ) and Y ∈ Γ(πM (u,t) is identified with (πM )∗(u,t) (X(u,t) ) ∈ Tu M , Y(·,t) is identified with (πM )∗ (Y(·,t) ) ∈ Γ(T M ) and Y(u,·) is identified with (πM )∗ (Yu,· ) ∈ C ∞ ([0, T ), Tu M ). Note that ∇ ∂ X = 0. Denote by the same sym∂t

∗ T (r,s) M defined in terms of ∇t ’s similarly. Define a bol ∇ the connection of πM H H connection ∇ of H by ∇X Y := (∇X Y )H for X ∈ Γ(M × [0, T )) and Y ∈ Γ(H). Similarly, define a connection ∇V of V by ∇VX Y := (∇X Y )V for X ∈ Γ(M × [0, T )) and Y ∈ Γ(V). Now we shall recall the evolution equations for some geometric quantities given in [Koi2]. By the same calculation as the proof of Lemma 4.2 of [Koi2] (where we replace H = kHkξ in the proof to H = hH, ξiξ = −H s ξ), we can derive the following evolution equation. ∗ (T (0,2) M ) satisfy the following evolution Lemma 5.1. The sections (gH )t ’s of πM equation: ∂gH = −2H s hH . ∂t

According to the proof of Lemma 4.3, we obtain the following evolution equation. Lemma 5.2. The unit normal vector fields ξt ’s satisfy the following evolution equation: ∂ξ = −F∗ (gradg H s ), ∂t ∗ (T M ) such that dH s (X) = g(grad H s , X) for where gradg (H s ) is the element of πM g ∗ any X ∈ πM (T M ). Let St (0 ≤ t < T ) be a C ∞ -family of a (r, s)-tensor fields on M and S a section ∗ (T (r,s) M ) defined by S ∗ (r,s) M ) of πM (u,t) := (St )u . We define a section △H S of πM (T by n X ∇ei ∇ei S, (△H S)(u,t) := i=1

∗ (T (r,s) M ) πM

∗ (T (r,s+1) M )) induced from ∇ and where ∇ is the connection of (or πM {e1 , · · · , en } is an orthonormal base of H(u,t) with respect to (gH )(u,t) . Also, we ¯ H SH of H(r,s) by define a section △

(△H H SH )(u,t)

:=

n X i=1

17

H ∇H ei ∇ei SH ,

where ∇H is the connection of H(r,s) (or H(r,s+1) ) induced from ∇H and {e1 , · · · , en } is as above. Let Aφ be the section of T ∗ V ⊗ T ∗ V ⊗ T V defined by e X Y e ) e (X, Y ∈ T V ). e X Y e ) e + (∇ AφX Y := (∇ e V H e H V H H

Also, let T φ be the section of T ∗ V ⊗ T ∗ V ⊗ T V defined by

e X Y e ) e (X, Y ∈ T V ). e X Y e ) e + (∇ TXφ Y := (∇ e V H e H V V V

Also, let At be the section of T ∗ M ⊗ T ∗ M ⊗ T M defined by

(At )X Y := (∇tXH YHt )Vt + (∇tXH YVt )Ht (X, Y ∈ T M ). t

t

∗ (T ∗ M πM

T ∗M

Also let A be the section of ⊗ ⊗ T M ) defined in terms of At ’s ∗ (t ∈ [0, T )). Also, let Tt be the section of T M ⊗ T ∗ M ⊗ T M defined by (Tt )X Y := (∇tXV YVt )Ht + (∇tXV YHt )Vt (X, Y ∈ T M ). t

Also let T be the section of [0, T )). Clearly we have

∗ (T ∗ M πM

t

⊗

T ∗M

⊗ T M ) defined in terms of Tt ’s (t ∈

F∗ (AX Y ) = AφF∗ X F∗ Y

for X, Y ∈ H and

F∗ (TW X) = TFφ∗ W F∗ X

for X ∈ H and W ∈ V. Let E be a vector bundle over M . For a section S of j k ∗ (T (0,r) M ⊗ E), we define Tr• S(· · · , •, πM · · · , •, · · · ) by gH (Tr•gH

j

k

S(· · · , •, · · · , •, · · · ))(u,t) =

n X i=1

j

k

S(u,t) (· · · , ei , · · · , ei , · · · )

((u, t) ∈ M ×[0, T )), where {e1 , · · · , en } is an orthonormal base of H(u,t) with respect j

k

to (gH )(u,t) , S(· · · , •, · · · , •, · · · ) means that • is entried into the j-th component and j

k

the k-th component of S and S(u,t) (· · · , ei , · · · , ei , · · · ) means that ei is entried into the j-th component and the k-th component of S(u,t) . In [Koi2], we derived the following relation. ∗ (T (0,2) M ) which is symmetric with respect Lemma 5.3. Let S be a section of πM to g. Then we have

(△H S)H (X, Y ) = (△H H SH )(X, Y ) −2Tr•gH ((∇• S)(A• X, Y )) − 2Tr•gH ((∇• S)(A• Y, X)) −Tr•gH S(A• (A• X), Y ) − Tr•gH S(A• (A• Y ), X) −Tr•gH S((∇• A)• X, Y ) − Tr•gH S((∇• A)• Y, X) −2Tr•gH S(A• X, A• Y ) 18

∗ (T (1,2) M ) induced from ∇. for X, Y ∈ H, where ∇ is the connection of πM

According to the proof of Lemma 4.5 in [Koi2], we obtain the following Simonstype identity. Lemma 5.4. We have △H h = ∇dH s + H s (A2 )♯ − (Tr (A2 )H )h, ∗ T (0,2) M ) defined by (A2 ) (X, Y ) := g(A2 X, Y ) where (A2 )♯ is the element of Γ(πM ♯ ∗ (X, Y ∈ πM T M ).

Note. In the sequel, we omit the notation F∗ for simplicity. ∗ (H(0,2) ) by Define a section R of πM

R(X, Y ) := Tr•gH h(A• (A• X), Y ) + Tr•gH h(A• (A• Y ), X) +Tr•gH h((∇• A)• X, Y ) + Tr•gH h((∇• A)• Y, X) +2Tr•gH (∇• h)(A• X, Y ) + 2Tr•gH (∇• h)(A• Y, X) (X, Y ∈ H). +2Tr•gH h(A• X, A• Y ) According to Theorem 4.6 in [Koi2], we obtain the following evolution equation from from Lemmas 5.2, 5.3 and 5.4. ∗ (T (0,2) M ) satisfies the following evolution Lemma 5.5. The sections (hH )t ’s of πM equation:

∂hH φ 2 s 2 s (X, Y ) = (△H H hH )(X, Y ) − 2H ((AH ) )♯ (X, Y ) − 2H ((Aξ ) )♯ (X, Y ) ∂t +Tr (AH )2 − ((Aφξ )2 )H hH (X, Y ) − R(X, Y )

for X, Y ∈ H.

According to the proof of Lemma 4.7, we obatin the following relation from Lemma 5.1. Lemma 5.6. Let X and Y be local sections of H such that g(X, Y ) is constant. Then we have g(∇ ∂ X, Y ) + g(X, ∇ ∂ Y ) = 2H s h(X, Y ). ∂t

∂t

According to Lemmas 4.8 and 4.10 in [Koi2], we obtain the following relation . 19

Lemma 5.7. For X, Y ∈ H, we have R(X, Y ) = 2Tr•gH h(Aφ• X, Aφ• (AH Y )i + h(Aφ• Y, Aφ• (AH X)i +2Tr•gH h(Aφ• X, AφY (AH •)i + h(Aφ• Y, AφX (AH •)i e • Aφ )ξ Y, Aφ• Xi + h(∇ e • Aφ )ξ X, Aφ• Y i (5.1) +2Tr•gH h(∇ e • Aφ )• X, Aφ Y i + h(∇ e • Aφ )• Y, Aφ Xi +Tr•gH h(∇ ξ ξ +2Tr•gH hT φφ ξ, Aφ• Y i, A• X

where we omit F∗ . In particular, we have R(X, X) = 4Tr•gH hAφ• X, Aφ• (AH X)i + 4Tr•gH hAφ• X, AφX (AH •)i e • Aφ )• X, Aφ Xi e • Aφ )ξ X, Aφ Xi + 2Tr• h(∇ +3Tr•gH h(∇ gH • ξ e X Aφ )ξ •i +Tr•gH hAφ• X, (∇

and hence

Tr•gH R(•, •) = 0.

Simple proof of the third relation. We give a simple proof of Tr•gH R(•, •) = 0. Take any (u, t) ∈ M × [0, T ) and an orthonormal base (e1 , · · · , en ) of H(u,t) with respect to g(u,t) . According to Lemma 5.3 and the definiton of R, we have (Tr•gH R(•, •))(u,t) = (Tr•gH (△H h)H (•, •))(u,t) − (Tr•gH (△H H hH )(•, •))(u,t) n X s (∇dH s )(ei , ei ) − (∇H dH s )(ei , ei ) = (△H H s )(u,t) − (△H H H )(u,t) =

n X (Aei ei )H s = 0, =−

i=1

i=1

where we use H s =

n P

h(ei , ei ) (which holds because the fibres of φ is regularized

i=1

minimal).

According to the proof of Corollary 4.11 in [Koi2], we obatin the following evolution equation. Lemma 5.8. The norms Hts ’s of Ht satisfy the following evolution equation: ∂H s = △H H s + H s ||AH ||2 − 3H s Tr((Aφξ )2 )H . ∂t 20

According to the proof of Corollary 4.12 in [Koi2], we obtain the following evolution equation. Lemma 5.9. The quantities ||(AH )t ||2 ’s satisfy the following evolution equation: ∂||AH ||2 = △H (||AH ||2 ) − 2||∇H AH ||2 ∂t +2||AH ||2 ||AH ||2 − Tr((Aφξ )2 )H −4H s Tr ((Aφξ )2 ) ◦ AH − 2Tr•gH R(AH •, •). From Lemmas 5.8 and 5.9, we obtain the following evolution equation. (H s )2

Lemma 5.10. The quantities ||(AH )t ||2 − nt ’s satisfy the following evolution equation: s 2 ∂(||(AH )t ||2 − (Hn ) ) 2 (H s )2 2 = △H ||AH || − + ||gradH s ||2 ∂t n n s )2 (H − 2||∇H AH ||2 +2||AH ||2 × ||AH ||2 − n (H s )2 φ 2 2 −2Tr((Aξ ) )H × ||AH || − n s H −4H s Tr (Aφξ )2 ◦ AH − id n Hs id •, • , −2Tr•gH R AH − n where gradH s is the gradient vector field of H s with respect to g and ||gradH s || is the norm of gradH s with respect to g. V H∗ the exterior product bundle of Set n := dim H = dim M and denote by nV ∗ ∗ degree n of H . Let dµgH be the section of πM ( n H∗ ) such that (dµgH )(u,t) is the volume element of (gH )(u,t) for any (u, t) ∈ M × [0, T ). Then we can derive the following evolution equation for {(dµgH )(·,t) }t∈[0,T ) . Lemma 5.11. The family {(dµgH )(·,t) }t∈[0,T ) satisfies ∂µgH = −(H s )2 · dµgH . ∂t 21

6

A maximum principle

Let M be a Hilbert manifold and gt (0 ≤ t < T ) a C ∞ -family of Riemannian metrics on M and G y M a almost free action which is isometric with respect to gt ’s (t ∈ [0, T )). Assume that the orbit space M/G is compact. Let Ht (0 ≤ t < T ) ∗ TM be the horizontal distribution of the G-action and define a subbundle H of πM by H(x,t) := (Ht )x . For a tangent vector field X on M (or an open set U of M ), we ¯ of π ∗ T M (or π ∗ T M |U ) by X ¯(x,t) := Xx ((x, t) ∈ M × [0, T )). define a section X M M t Let ∇ (0 ≤ t < T ) be the Riemannian connection of gt and ∇ the connection of ∗ T M defined in terms of ∇t ’s (t ∈ [0, T )). Define a connection ∇H of H by ∇H Y = πM X ∗ T (r0 ,s0 ) M ), prH (∇X Y ) for any X ∈ T (M × [0, T )) and any Y ∈ Γ(H). For B ∈ Γ(πM ∗ T (r,s) M ) to Γ(π ∗ T (r+r0 ,s+s0 ) M ) by we define maps ψB⊗ and ψ⊗B from Γ(πM M ∗ ψB⊗ (S) := B ⊗ S, and ψ⊗B (S) := S ⊗ B (S ∈ Γ(πM T (r,s) M ),

∗ T (r,s) M ) to Γ(π ∗ T (kr,ks) M ) by respectively. Also, we define a map ψ⊗k of Γ(πM M ∗ ψ⊗k (S) := S ⊗ · · · ⊗ S (k−times) (S ∈ Γ(πM T (r,s) M ).

∗ T (0,s) M ) (or Γ(π ∗ T (1,s) M )) to Also, we define a map ψgH ,ij (i < j) from Γ(πM M ∗ T (0,s−2) M ) (or Γ(π ∗ T (1,s−2) M )) by Γ(πM M

(ψgH ,ij (S))(x,t) (X1 , · · · , Xs−2 ) n X := S(x,t) (X1 , · · · , Xi−1 , ek , Xi+1 , · · · , Xj−1 , ek , Xj+1 , · · · , Xs−2 ) k=1

∗ T (1,s) M ) to Γ(π ∗ T (0,s−1) M ) by and define a map ψH,i from Γ(πM M

(ψH,i (S))(x,t) (X1 , · · · , Xs−1 ) := Tr S(x,t) (X1 , · · · , Xi−1 , •, Xi , · · · , Xs−1 ), where Xi ∈ Tx M (i = 1, · · · , s − 1) and {e1 , · · · , en } is an orthonormal base of (Ht )x ∗ T (0,s) M ) to oneself given by the with respect to gt . We call a map Pg from Γ(πM composition of the above maps of five type a map of polynomial type. In [Koi2], we proved the following maximum principle for a C ∞ -family of Ginvariant (0, 2)-tensor fields on M . ∗ (T (0,2) M )) such that, for each t ∈ [0, T ), S (:= S Theorem 6.1. Let S ∈ Γ(πM t (·,t) ) is a G-invariant symmetric (0, 2)-tensor field on M . Assume that St ’s (0 ≤ t < T ) satisfy the following evolution equation:

(7.1)

∂SH H = △H ¯ 0 SH + Pg (S)H , H SH + ∇X ∂t 22

∗ (T (0,2) M )) to where X0 ∈ Γ(T M ) and Pg is a map of polynomial type from Γ(πM oneself. (i) Assume that Pg satisfies the following condition:

X ∈ Ker(SH + εgH )(x,t) ⇒ Pg (SH + εgH )(x,t) (X, X) ≥ 0 for any ε > 0 and any (x, t) ∈ M × [0, T ). Then, if (SH )(·,0) ≥ 0 (resp. > 0), then (SH )(·,t) ≥ 0 (resp. > 0) holds for all t ∈ [0, T ). (ii) Assume that Pg satisfies the following condition: X ∈ Ker(SH + εgH )(x,t) ⇒ Pg (SH + εgH )(x,t) (X, X) ≤ 0 for any ε > 0 and any (x, t) ∈ M × [0, T ). Then, if (SH )(·,0) ≤ 0 (resp. < 0), then (SH )(·,t) ≤ 0 (resp. < 0) holds for all t ∈ [0, T ). Similarly we obtain the following maximal principle for a C ∞ -family of G-invariant functions on M . Theorem 6.2. Let ρ be a C ∞ -function over M ×[0, T ) such that, for each t ∈ [0, T ), ρt (:= ρ(·, t)) is a G-invariant function on M . Assume that ρt ’s (0 ≤ t < T ) satisfy the following evolution equation: ∂ρ ¯0 ) + P ◦ ρ, = △H ρ + dρ(X ∂t where X0 ∈ Γ(T M ) and P is a polynomial function over R. (i) Assume that P satisfies P (0) ≥ 0. Then, if ρ0 ≥ 0 (resp. > 0), then ρt ≥ 0 (resp. > 0) holds for all t ∈ [0, T ). (ii) Assume that P satisfies P (0) ≥ 0. Then, if ρ0 ≤ 0 (resp. < 0), then ρt ≤ 0 (resp. < 0) holds for all t ∈ [0, T ).

7

Sobolev inequality for Riemannian suborbifolds

In this section, we prove the divergence theorem for a compact Riemannian orbifold and Sobolev inequality for a compact Riemannian suborbifold, which may have the boundary. Let (M , g) be an n-dimensional compact Riemannian orbifold, Σ the singular set of (M , g) and {Σ1 , · · · , Σk } be the set of Z all connected components of Σ. ′

ρ dvg the integral of ρ over Set M := M \ Σ. For a function ρ over M , we call ′ M Z M and denote it by ρ dvg . First we prove the divergence theorem for orbitangent M

vector fields on a compact Riemannian orbifold. 23

Theorem 7.1. For any C 1 -orbitangent vector field X on (M , g), the relation Z divg X dvg = 0 holds. M

Proof. Let U i (i = 1, · · · , k) be a sufficiently small tubular neighborhood of Σi with k bij /Γij ) | j = U i ∩ U j = ∅ (i 6= j). Set W := M \ ∪ U i . Take families {(Uij , ϕij , U i=1

i 1, · · · , mi } of orbifold charts of M such that the family {cl(Uij )}m j=1 of the clusure cl(Uij ) of Uij gives a division of cl(U i ) (i = 1, · · · , k). Denote by πij the projection bij → U bij /Γij and li the cardinal number of Γij , which depends only on πΓij : U i. Let ξi is the outward unit normal vector field of ∂Ui and ιi is the inclusion map of ∂U i into M . Also, let ξij be the outward unit normal vector field of ∂Uij satisfying ξij |∂U i ∩∂Uij = ξi |∂C i ∩∂Uij . Also, let ξbij be the unit normal vector field bij be the vector field on U bij bij satisfying (ϕ−1 ◦ πij )∗ (ξbij ) = ξij |∂U and let X of ∂ U ij ij −1 bij ) = X|U . Denote by ιij the inclusion map of ∂ U bij into satisfying (ϕ ◦ πij )∗ (X

ij

ij

bij /Γij ). Then, by using the and b g ij the local lift of g with respect to (Uij , ϕij , U divergence theorem (for a compact Riemannian manifold with boundary), we have

Rn

(7.1)

Z

divg X dvg =

M

Z

divg X dvg +

W

=−

k Z X i=1

mi Z k X X i=1 j=1

g(X, ξi ) dv

ι∗i g

∂Ui

divg X dvg Uij

+

mi Z k X X i=1 j=1

divg X dvg . Uij

Also, by using the divergence theorem, we can show Z Z Z 1 1 b bij , ξbij ) dv ∗ b b divg X dvg = divbg Xij dvbg = g ij (X ιij g ij ij ij li Ubij l bij i Uij ∂ U Z =

g(X, ξij ) dvg .

∂Uij

and hence (7.2)

mi Z X j=1

divg X dvg = Uij

From (7.1) and (7.2), we obtain

Z

Z

∂Ui

g(X, ξi ) dvι∗i g .

divg X dvg = 0. M

24

’s : ξj

Ui

in fact

’s : ξi

Σi ’s : ξij

Σi

Uij Σi

Ui

Ui

Figure 4 : The relation of ξi and ξij

In 1973, J.H. Michael and L.M. Simon ([MS]) proved the Sobolev inequality for compact Riemannian submanifolds (which may have boundary) in a Euclidean space, where we note that the integrand must vanishes on the boundary of the submanifold. In 1974, D. Hoffman and J. Spruck ([HS]) proved the same Sobolev inequality in a general Riemannian manifold, where we note that the integrand must vanishes on the boundary of the submanifold and furthemore, the volume of the support of the integrand must satisfy some estimate from above related to the curvature and the injective radius of the ambient space. We shall show the following Sobolev inequality for compact Riemannian suborbifolds. Theorem 7.2. Let (M , g) be a compact Riemannian suborbifold isometrically immersed into (N, gN ) by f . Assume that the sectional curvature K of a complete Riemannian orbifold (N, gN ) satisfies K ≤ b2 (b is a non-negative real number or the purely imaginary number), M satisfies the following condition (∗1 ): (∗1 ) f (M ) is included by B πb (x0 ) for some x0 ∈ Σ and expx0 |B Tπ (x0 ) is injective. b

Let ρ be any non-negative (∗2 )

C 1 -function

b2 (1 − α)−2/n

on M satisfying 2/n ≤ 1, ωn−1 · l · Volg (supp ρ)

where g denotes the induced metric on M , ωn denotes the volume of the unit ball in the Euclidean space Rn , l denotes the cardinality of the local group at x0 and α is any fixed positive constant smaller than one. Then the following inequality for ρ holds: n−1 Z Z n n ≤ C(n) ρ n−1 dvg (||dρ|| + ρ · |H s |) dvg , (7.3) M

M

25

where H s denotes the mean curvature of f and C(n) is the positive constant depending only on n. Remark 7.1. In the case where (M , g) is a compact Riemannian manifold, the statement of this theorem follows from the Sobolev’s inequality in [HS] because the condition (∗1 ) assures the condition (2.3) in Theorem 2.1 of [HS]. We shall prepare some lemmas to prove this theorem. Let Γ be the local group at x0 and set x b0 := (expx0 ◦π)−1 (x0 ). The orbitangent space Tx0 M is identified with b π (b the orbit space Rn+1 /Γ. Let π : Rn+1 → Tx0 N be the orbit map. Set B x0 ) := b −1 ∗ (expx0 ◦π) (B πb (x0 )) and gc N := (expx0 ◦π) gN . Also, set c := {(x, yb) ∈ M × B b π (b x0 ) | f (x) = (expx0 ◦π)(b y )} M b

c ֒→ B c ). Also, define a map b π (b and define a map fb : M x0 ) by fb(x, yb) = yb ((x, yb) ∈ M b (x, yb6 )

patch

c M

(x, yb5 )

(x, yb4 )

(x, yb3 )

(x, yb2 )

yb4

fb

yb5

(x, yb1 )

B πb (x0 ) f (M )

πM M

yb3

x b0

yb1

yb2

π Tx0 N

Rn+1 yb6

expx0 x0

x

f

b π (b B x0 ) b

in fact

N x0

etc. edge cone-case

Figure 5 : The lift of an orbisubmanifold in a singular geodesic ball (I)

26

(x, yb2 )

πc M (x, yb3 )

(x, yb1 ) c M

(x, yb4 )

yb2 fb

yb3

yb1

x b0

Rn+1

yb4

π

πM

b π (b x0 ) B b

Tx0 N expx0 B πb (x0 ) x0

x

singular point

f

M

N

Figure 6 : The lift of an orbisubmanifold in a singular geodesic ball (II) c → M by π (x, yb) := x ((x, yb) ∈ M c ). It is clear that M c is a C ∞ -manifold πM : M M and that fb is a C ∞ -immersion. Also, it is clear that πM is a C ∞ -orbisubmersion c and that expx0 ◦π ◦ fb = f ◦ πM holds. Let b g be the Riemannian metric on M c, b d N be the g) → (M , g) is a Riemannian orbisubmersion. Let ∇ such that π : (M M

b b π (b Riemannian connection of gc x0 ))), define g. For X ∈ Γ∞ (fb∗ T (B N and ∇ that of b b c ) by div X ∈ C ∞ (M fb

d N )fbX)) = (divfbX)xb := (Trbg ((∇ x b

n X i=1

c ), d N )fb X, fb (e )) (b b g ((∇ x∈M ∗b x i ei

c with respect to b d N )fb where (e1 , · · · , en ) is an orthonormal base of Txb M g xb and (∇ d N by fb. Let X T (resp. X ⊥ ) be the tangential denotes the induced connection of ∇ (resp. the normal) component of X, that is, Xxb = f∗bx (XxbT ) + Xxb⊥

c ). c, X ⊥ ∈ T ⊥M (XxbT ∈ Txb M x b x b

27

First we prepare the following lemma. Lemma 7.3. (i) divbg X T = divfbX + gc N (X, H), where H denotes the mean curvature vector of fb. c ), we have (ii) For ρ ∈ C ∞ (M g (X T , gradbg ρ), divfb(ρX) = ρdivfbX + b

g. where gradbg (•) denotes the gradient vector field of (•) with respect to b

See the proof of Lemma 3.2 in [HS] about the proof of this lemma. Let rxb0 : b π (b B x0 ) → [0, ∞) be the distance function from x b0 with respect to gc N , that is, b b π (b b π (b rxb0 (b x) := dgc (b x0 , x b) (b x∈B x0 )). Define a C ∞ -vector field P on B x0 ) by N b b rb0 )xb Pxb := rxb0 (b x) · (gradgc N x

b π (b (b x∈B x0 )), b

(•) denotes the gradient vector field of • with respect to gc where gradgc N . Also, N ∞ n+1 e over the tangent space Txb R define a C -vector field P by 0 e v := v P

b π (b B x0 ) b

(v ∈ Txb0 Rn+1 ).

x b0

’s : P Rn+1

π Tx0 N expx0

B πb (x0 )

N

Figure 7 : The position vector field on the orbicovering of a geodesic ball By using the discussion in the proof Lemmas 3.5 and 3.6 in [HS], we can show the following fact. 28

b π (b b π (b Lemma 7.4. (i) For any unit vector v of B x0 ) at any x b∈B x0 ), the following b b inequality holds: d N gc x) · cot(b · rxb0 (b x)). N (∇ v P, v) ≥ b · rx b0 (b c , the following inequality holds: (ii) For any (x, yb) ∈ M

(divfb(P ◦ fb))(x,by ) ≥ n · b · rxb0 (b y ) · cot(b · rxb0 (b y )).

Let λ be a C 1 -function over R satisfying the following condition: (7.4)

λ′ (t) ≥ 0 (t ≥ 0).

λ(t) = 0 (t ≤ 0),

c bM bs (b bs (b Set B x0 ) := fb−1 (B x0 )), where B x0 ) denotes the geodesic ball of radius s s (b 1 centered at x b0 . Let ρ be a C -function as in the statement of Theorem 7.2 and set ρb := ρ ◦ πM . Define functions φρb,bx0 ,λ , ηρb,bx0 , φρb,bx0 , η ρb,bx0 over [0, πb ) by Z λ(s − (rxb0 ◦ fb)) · ρb dvbg , φρb,bx0 ,λ (s) := c M Z λ(s − (rxb0 ◦ fb)) · (|gradbg ρ| + ρ|H|) dvbg , ηρb,bx0 ,λ (s) := c M Z φρb,bx0 (s) := c ρb dvbg , bsM (b B x0 ) Z λ(s − (rxb0 ◦ fb)) · (|gradbg ρ| + ρ|H|) dvbg . η ρb,bx0 ,λ (s) := c bsM (b x0 ) B

According to the proof of Lemma 4.1 in [HS], we can derive the following fact by using Lemmas 7.3 and 7.4. Lemma 7.5. For all s ∈ [0, πb ), the following inequality hold:    − d (sin(bs))−n φρb,bx0 ,λ (s) ≤ (sin(bs))−n ηρb,bx0 ,λ (s) (b : real) ds d −n   − s φρb,bx0 ,λ (s) ≤ s−n ηρb,bx0 ,λ (s) (b : purely imaginary). ds

According to the proof of Lemma 4.2 in [HS], we can show the following result by using Lemma 7.5.

29

Lemma 7.6. Let α and α ˆ be constants with 0 < α < 1 ≤ α ˆ , and {λε }ε>0 be C 1 functions over R satisfying the above condition (7.4) and the following condition: λε ≤ 1,

(7.5)

λ−1 ε (1) = [ε, ∞).

Assume that the following conditions hold: ρb(b x0 ) ≥ 1,

(i) Set

(ii)

b

2

1 (1 − α)ωn

Z

c M

ρb dvbg

2

n

≤ 1.

(  1) Z n  1 1   (b : real)  b · arcsin b (1 − α)ω c ρ dvbg n M sρb := 1 Z  n  1   (b : purely imaginary). ρb dvbg c (1 − α)ωn M

which is defined by the above condition (ii). Furthermore, assume the following: α ˆ sρ ≤

(iii)

π . b

Then there exists s1 ∈ (0, sρ ) satisfying φρb,bx0 (ˆ αs1 ) ≤ α−1 · α ˆ n−2 · sρ · lim η ρb,bx0 ,λε (s1 ). ε→0

By using Lemma 7.6, we prove Theorem 7.2. Proof of Theorem 7.2. We shall prove the statement in the case where b is real (similar also the case where b is purely imaginary). Let α, α ˆ be constants with 0 0) over R by λε (s) := λε (s + ε) and define a function c by ρb := λ (b ρb (ε > 0) over M ρ − t). Since ρ satisfies the condition (∗ ) in ε,t

ε,t

ε

2

Theorem 7.2, ρbε,t satisfies the conditions (ii) and (iii) in Lemma 7.6. By using Lemma 7.6 and discussing as in the proof of Theorem 2.1 in [HS], we can derive Z

c M

ρb

n n−1

dvbg

n−1 n

≤

1 n π −1 · · α−1 · α ˆ n−2 · (1 − α)− n · ωn n n− Z 12 kgradbg ρbk + ρb · kH s k dvbg . ×

c M

30

Clearly we have

Z

c M

and

Z

c M

ρb

n n−1

dvbg = l ·

M

ρ M

n n−1

dvg

n−1 n

n

ρ n−1 dvg

M

Z s kgradbg ρbk + ρb · kH k dvbg = l ·

Hence we obtain Z

Z

kgradg ρk + ρ · kH s k dvg .

1 1 π n −1 · · α−1 · α ˆ n−2 · (1 − α)− n · ωn n ≤ ln · Zn−1 2 × kgradg ρk + ρ · kH s k dvg .

M

8

Approach to horizontally totally umbilicity

In this section, we recall the preservability of horizontally strongly convexity along the mean curvature flow. Let G y V be an isometric almost free action with minimal regularizable orbit of a Hilbert Lie group G on a Hilbert space V equipped with an e the inner product h , i and φ : V → N := V /G the orbit map. Denote by ∇ Riemannian connection of V . Set n := dim N − 1. Let M (⊂ V ) be a G-invariant hypersurface in V such that φ(M ) is compact. Let f be an inclusion map of M into V and ft (0 ≤ t < T ) the regularized mean curvature flow starting from f . We use the notations in Sections 5. In the sequel, we omit the notation ft∗ for simplicity. As stated in Introduction, set (♯)

L := sup

max

e1 )5u u∈V (X1 ,··· ,X5 )∈(H

e X Aφ )X X4 ), X5 i|, |hAφX1 ((∇ 2 3

e1 := {X ∈ H e | ||X|| = 1}. Assume that L < ∞. Note that L < ∞ in the where H case where N is compact. In [Koi2], we proved the following horizontally strongly convexity preservability theorem by using evolution equations stated in Section 3 and the discussion in the proof of Theorem 6.1. Theorem 8.1([Koi2]). If M satisfies (H0s )2 (hH )(·,0) > 2n2 L(gH )(·,0) , then T < ∞ holds and (Hts )2 (hH )(·,t) > 2n2 L(gH )(·,t) holds for all t ∈ [0, T ). In this section and the next section, we shall derive some important facts to prove the statement (ii) of this theorem. 31

Let f and ft (0 ≤ t < T ) be as in the statement of Thoerem A. Then, according to Theorem 8.1, for all t ∈ [0, T ), (8.1)

(Hts )2 (hH )(·,t) > 2n2 L(gH )(·,t)

holds. In this section, we shall prove the following result for the approach to the horizontally totally umbilicity of ft as t → T . Proposition 8.2. Under the hypothesis of Theorem 8.1, there exist positive constants δ and C0 depending on only f, L, K and the injective radius i(N ) of N such that (Hts )2 < C0 (Hts )2−δ ||(AH )(·,t) ||2 − n holds for all t ∈ [0, T ). We prepare some lemmas to show this proposition. In the sequel, we denote (r,s) induced from g by the same symbol g , and set ||S|| := the H H p fibre metric of H (r,s) gH (S, S) for S ∈ Γ(H ). Define a function ψδ over M by (H s )2 1 2 ||AH || − . ψδ := (H s )2−δ n Lemma 8.2.1. Set α := 2 − δ. Then we have ∂ψδ (α − 1)(α − 2) = △H ψδ + (2 − α)||AH ||2 ψδ + || dH s ||2 ψδ ∂t (H s )2 2 2(α − 1) s ||H||∇H AH − dH s ⊗ AH 2 g (dH , dψ ) − + H δ Hs (H s )α+2 6 φ 2 +3(α − 2)Tr((Aξ ) )H )ψδ − Tr((Aφξ )2 ◦ AH ) s α−1 (H ) 6 4 φ 2 + Tr((Aξ ) ) − Tr· Tr• h(((∇• A)• ◦ AH )·, ·) n(H s )α−2 (H s )α gH gH 4 4 − s α Tr·gH Tr•gH h((A• ◦ AH )·, A• ·) + Tr· Tr• h((A• ◦ A• )·, AH ·). (H ) (H s )α gH gH

32

Proof. By using Lemmas 5.8 and 5.10, we have 1 ∂ψδ = (2 − α)||AH ||2 ψδ + △ (||AH ||2 ) s )α H ∂t (H (α − 2)(H s )2 1 2 △H H s − s α+1 α||AH || − (H ) n 2 − s α ||∇H AH ||2 + (3α − 2)Tr((Aφξ )2 )H · ψδ (H ) 6 Hs φ 2 − s α−1 Tr (Aξ ) ◦ (AH − id) (H ) n 4 − s α Tr·gH Tr•gH h((A• ◦ AH )·, A• ·) (H ) 4 + s α Tr·gH Tr•gH h((A• ◦ A• )·, AH ·) (H ) 4 − s α Tr·gH Tr•gH h((∇• A)• ◦ AH )·, ·). (H )

(8.2)

Also we have 1 2α △H ||AH ||2 − gH (dH s , d(||AH ||2 )) s α (H ) (H s )α+1 1 (α − 2)(H s )2 2 − s α+1 α||AH || − △H H s (H ) n (α − 1)(α − 2)(H s )2 1 2 ||dH s ||2 + s α+2 α(α + 1)||AH || − (H ) n

△H ψδ = (8.3)

From (8.2) and (8.3), we obtain the desired relation. Then we have the following inequalities. By using the Codazzi equation, we can derive the following relation. Lemma 8.2.2. For any X, Y, Z ∈ H, we have H (∇H X hH )(Y, Z) = (∇Y hH )(X, Z) + 2h(AX Y, Z) − h(AY Z, X) + h(AX Z, Y )

or equivalently, H (∇H X AH )(Y ) = (∇Y AH )(X) + 2(A ◦ AX )Y + (AY ◦ A)(X) − (AX ◦ A)(Y ).

Proof. Let (x, t) be the base point of X, Y and Z and extend these vectors to sections e Ye and Ze of Ht with (∇H X) e (x,t) = (∇H Ye )(x,t) = (∇H Z) e (x,t) = 0. Since ∇h is X, 33

symmetric with respect to g by the Codazzi equation and the flatness of V , we have e e (∇H X hH )(Y, Z) = X(h(Y , Z)) = (∇X h)(Y, Z) + h(AX Y, Z) + h(AX Z, Y ) = (∇Y h)(X, Z) + h(AX Y, Z) + h(AX Z, Y ) e Z)) e − h(AY X, Z) − h(AY Z, X) + h(AX Y, Z) + h(AX Z, Y ) = Y (h(X, H = (∇Y h)(X, Z) + 2h(AX Y, Z) − h(AY Z, X) + h(AX Z, Y ). Set K :=

max

(e1 ,e2 ):o.n.s. of T V

||Aφe1 e2 ||2 ,

where ”o.n.s.” means ”orthonormal system”. Assume that K < ∞. Note that K < ∞ if N = V /G is compact. For a section S of H(r,s) and a permutaion σ of s-symbols, we define a section Sσ of H(r,s) by Sσ (X1 , · · · , Xs ) := S(Xσ(1) , · · · , Xσ(s) ) (X1 , · · · , Xs ∈ H) and Alt(S) by Alt(S) :=

1 X sgn σ Sσ , s! σ

where σ runs over the symmetric group of degree s. Also, denote by (i, j) the transposition exchanging i and j. Since (Hts )2 (hH )(·,t) > n2 L(gH )(·,t) (t ∈ [0, T )) and φ(M ) is compact, there exists a positive constant ε satisfying (♯)

||H(·,0) ||2 (hH )(·,0) ≥ n2 L(gH )(·,0) + ε||H(·,0) ||3 (gH )(·,0) .

Then we can show that ||H(·,t) ||2 (hH )(·,t) ≥ n2 L(gH )(·,t) + ε||H(·,t) ||3 (gH )(·,t) holds for all t ∈ [0, T ). Without loss of generality, we may assume that ε ≤ 1. Then we have the following inequalities. Lemma 8.2.3. Let ε be as above. Then we have the following inequalities: (H s )2 , (8.4) H s TrH (AH )3 − ||(AH )t ||4 ≥ nε2 (H s )2 ||(AH )t ||2 − n and (8.5)

s H H ∇ AH − dH s ⊗ AH 2 1 s s )2 . ≥ −8ε−2 K(H(u,t) )2 + ||(dH s )(u,t) ||2 ε2 (H(u,t) 8 34

Proof. First we shall show the inequality (8.4). Fix (u, t) ∈ M × [0, T ). Take an orthonormal base {e1 , · · · , en } of H(u,t) with respect to g(u,t) consisting of the eigenvectors of (AH )(u,t) . Let (AH )(u,t) (ei ) = λi ei (i = 1, · · · , n). Note that λi > εH s (> 0) (i = 1, · · · , n). Then we have X X (λi − λj )2 . H s TrH (AH )3 − ||(AH )t ||4 = λi λj (λi − λj )2 > ε2 (H s )2 1≤i 1. From (8.30) and this inequality, we obtain Z T Z 1/q0 2q0 ˆ ψB (·, t) d¯ vt dt 0 M (8.31) 1/r Z T Z r−1 2r s 2r −1 r ˆ ˆ . ψB (Ht )B d¯ vt dt ≤ 2C(n, k) βδ||At (k)||T · 0

M

On the other hand, according to Lemma 8.2.10, we have Z 2r (8.32) ψbB (Hts )2r vt ≤ C 2r B d¯ M

for some positive constant C (depending only on K, L and f ) by replacing r to a bigger positive number if necessary. Also, by using the Hölder inequality, we obtain Z T Z 2 ψˆB (·, t) d¯ vt dt M 0 Z T Z 1/q0 q0 −1 q0 2q0 ˆ ≤ ||At (k)||T ψB (·, t) d¯ vt dt . · 0

M

From (8.31), (8.32) and this inequality, we obtain Z T Z 2−1/q0 −1/r 2 ˆ k)−1 · 2βδ. ψˆB (·, t) d¯ vt dt ≤ ||At (k)||T · C 2 · C(n, (8.33) 0

M

We may assume that 2 − 1/q0 − 1/r > 1 holds by replacing r to a bigger positive number if necessary. Take any positive constants h and k with h > k ≥ k1 . Then 48

we have

Z

T 0

≥

Z

Z T

0

M

β ψδ,k dv t

Z

At (h)

dt ≥

Z

T

Z

0!

|h − k|β dv t

M

β

(ψδ,k − ψδ,h ) dv t dt

dt = |h − k|β · ||At (h)||T .

From this inequality and (8.33), we obtain (8.34)

2−1/q0 −1/r

|h − k|β · ||At (h)||T ≤ ||At (k)||T

ˆ k)−1 · 2βδ. · C 2 · C(n,

Since • 7→ ||At (•)||T is a non-increasing and non-negative function and (8.34) holds for any h > k ≥ k1 , it follows from the Stambaccha’s iteration lemma that ||At (k1 + d)||T = 0, where d is a positive constant depending only on β, δ, q0 , r, C, Cb (n, k) and ||At (k1 )||T . This implies that sup max ψδ (·, t) ≤ k1 + d < ∞. This completes the t∈[0,T )

M

proof.

9

Estimate of the gradient of the mean curvature from above

In this section, we shall derive the following estimate of gradH s from above by using Proposition 8.2. Proposition 9.1. For any positive constant b, there exists a constant C(b, f0 ) (depending only on b and f0 ) satisfying ||grad H s ||2 ≤ b · (H s )4 + C(b, f0 ) on M × [0, T ).

We prepare some lemmas to prove this proposition. Lemma 9.1.1. The family {||gradt Hts ||2 }t∈[0,T ) satisfies the following equation:

(9.1)

∂||grad H s ||2 − △H (||grad H s ||2 ) ∂t = −2||∇H grad H s ||2 + 2||AH ||2 · ||grad H s ||2 +2H s · gH (grad(||AH ||2 ), grad H s ) +2gH ((AH )2 (grad H s ), grad H s ) −6H s · gH (grad(Tr((Aφξ )2 )H , grad H s ) −6Tr((Aφξ )2 )H · ||grad H s ||2 . 49

Hence we have the following inequality: ∂||grad H s ||2 − △H (||grad H s ||2 ) ∂t ≤ −2||∇H grad H s ||2 + 4||AH ||2 · ||grad H s ||2 +2H s · gH (grad(||AH ||2 ), grad H s ) +6H s · ||grad(Tr((Aφξ )2 )H )|| · ||grad H s || −6Tr((Aφξ )2 )H · ||grad H s ||2 .

(9.2)

Proof. By using Lemmas 5.1 and 5.8, we have ∂||grad H s ||2 ∂gH ∂H s s s s = (grad H , grad H ) + 2gH grad , grad H ∂t ∂t ∂t = −2H s · hH (grad H s , grad H s ) + 2gH (grad(△H H s ), grad H s ) +2gH (grad(H s · ||AH ||2 )), grad H s ) −6gH (grad(H s · Tr((Aφξ )2 )H , grad H s ). Also we have s s △H (||grad H s ||2 ) = 2gH (△H H (grad H ), grad H ) +2gH (∇H grad H s , ∇H grad H s )

and s s s s 2 s △H H (grad H ) = grad(△H H ) + H · AH (grad H ) − (AH ) (grad H ).

By using these relations and noticing gH (AH (•), ·) = −hH (•, ·), we can derive the desired evolution equation (9.1). The inequality (9.2) is derived from (9.1) and gH ((AH )2 (grad H s ), grad H s ) ≤ ||AH ||2 · ||grad H s ||2 .

Lemma 9.1.2. The family

(9.3)

||gradt Hts ||2 Hts

satisfies the following inequality:

t∈[0,T )

||grad H s ||2 ∂ ||grad H s ||2 − △H ∂t Hs Hs s 2 3||grad H || · ||AH ||2 + 2gH (grad(||AH ||2 , grad H s ) ≤ Hs 3 +6||grad(Tr((Aφξ )2 )H || · ||grad H s || − s Tr((Aφξ )2 )H · ||grad H s ||2 H 50

Proof. By a simple calculation, we have ∂ ||grad H s ||2 ||grad H s ||2 − △H ∂t Hs Hs s 2 ∂||grad H || 1 s 2 − △H (||grad H || ) = s H ∂t ||grad ||H|| ||2 ∂H s s − − △ H H (H s )2 ∂t 2 + s 2 gH (grad H s , grad(||grad H s ||2 )). (H ) From this relation, Lemmas 5.8 and (9.2), we can derive the desired inequality. From Lemma 5.8, we can derive the following evolution equation directly. Lemma 9.1.3. The family (Hts )3 t∈[0,T ) satisfies the following evolution equation: ∂(H s )3 − △H ((H s )3 ) ∂t = 3(H s )3 · ||AH ||2 − 6H s · ||grad H s ||2 − 9(H s )3 · Tr((Aφξ )2 )H .

By using Lemmas 5.8, 5.10 and Proposition 8.2, we can derive the following evolution inequality. Lemma 9.1.4. The family

||(AH )t ||2 −

(Hts )2 n

· Hts

satisfies the follow-

t∈[0,T )

ing evolution inequality: (H s )2 (H s )2 ∂ 2 s s 2 ||AH || − · H − △H ·H ||AH || − ∂t n n 2(n − 1) s ˘ C0 , δ) · ||∇H AH ||2 H · ||∇H AH ||2 + C(n, ≤− 3n (H s )2 (H s )2 φ 2 s 2 2 s 2 +3H · ||AH || · ||AH || − − 2H · Tr((Aξ ) )H · ||AH || − n n s H Hs φ 2 s 2 s • −4(H ) · Tr (Aξ ) ) ◦ AH − · id − 2H · TrgH R AH − · id (•), • n n (H s )2 −3 ||AH ||2 − · H s · Tr((Aφξ )2 )H . n 51

Proof. By using Lemmas 5.8 and 5.10, we can derive ∂ (H s )2 (H s )2 2 s s 2 ||AH || − · H − △H ·H ||AH || − ∂t n n (H s )2 2H s · ||grad H s ||2 + 2H s · ||AH ||2 · Tr(((AH )2 ) − = n n (H s )2 s H 2 s 2 2 −2H · ||∇ AH || + H · ||AH || · ||AH || − n s H −gH grad ||AH ||2 − · id , grad H s n (9.4) (H s )2 φ 2 s 2 −2H · Tr((Aξ ) )H · Tr(((AH ) ) − n Hs φ 2 s 2 −4(H ) · Tr (Aξ ) ) ◦ AH − · id n Hs s • · id (•), • AH − −2H · TrgH R n (H s )2 −3 ||AH ||2 − · H s · Tr((Aφξ )2 )H . n s 2

s

On the othe hand, by using ||AH ||2 − (Hn ) = ||AH − Hn · id ||, we can derive s )2 (H s 2 gH grad ||AH || − , grad H n s )2 (H s 2 (grad H ) = d ||AH || − n Hs Hs H = 2 gH ∇gradH s AH − · id , AH − · id n n s H · id ≤ 2||gradH s || · ||∇H AH || · AH − n Hs H 2 ≤ 2n||∇ AH || · AH − · id , n

where we use have

Hence we have (9.5)

1 s 2 n ||gradH ||

≤ ||∇H AH ||2 . Also, according to Proposition 8.2, we s p AH − H · id ≤ C0 · (H s )1−δ/2 . n

s 2 gH grad ||AH ||2 − (H ) , grad H s n p ≤ 2n C0 ||∇H AH ||2 · (H s )1−δ/2 . 52

Furthermore, according to the Young’s inequality: ab ≤ ε · ap + ε−1/(p−1) · bq (∀ a > 0, b > 0)

(9.6).

(where p and q are any positive constants with constant), we have (9.7)

2n

p

C0 (H s )1−δ/2 ≤

1 p

+

1 q

= 1 and ε is any positive

2(n − 1) ˘ C0 , δ), · H s + C(n, 3n

˘ C0 , δ) is a positive constant only on n, C0 and δ. Also, we have where C(n, ||∇H AH ||2 ≥

3 ||∇H H||2 . n+2

From (9.4) and these inequalities, we can derive the desired evolution inequality. By using Lemmas 5.9, 9.1.2, 9.1.3 and 9.1.4, we shall prove Theorem Proposition 9.1. Proof of Proposition 9.1. Define a function ρ over M × [0, T ) by ||grad H s ||2 (H s )2 s 2 ˘ C0 , δ)||AH ||2 − b(H s )3 , ρ := + C1 H ||AH || − + C1 · C(n, Hs n where b is any positive constant and C1 is a positive constant which is sufficiently

53

big compared to n and b. By using Lemmas 5.9, 9.1.2, 9.1.3 and 9.1.4, we can derive

(9.8)

∂ρ − △H ρ ∂t 3||grad H s ||2 · ||AH ||2 + 2gH (grad(||AH ||2 ), grad H s ) ≤ Hs 2(n − 1) · C1 · H s · ||∇H AH ||2 − 3n (H s )2 +3C1 · H s · ||AH ||2 ||AH ||2 − n 4 ˘ +2C1 · C(n, C0 , δ) · ||AH || − 3b(H s )3 · ||AH ||2 + 6bH s · ||grad H s ||2 3 +6||grad H s || · ||grad(Tr((Aφξ )2 )H )|| − s · ||grad H s ||2 · Tr((Aφξ )2 )H H (H s )2 φ 2 s 2 −2C1 H · Tr((Aξ ) )H · ||AH || − n s H φ −4C1 (H s )2 · Tr (Aξ )2 ) ◦ AH − · id n s H s • · id (•), • −2C1 H · TrgH R AH − n (H s )2 · H s · Tr((Aφξ )2 )H −3C1 ||AH ||2 − n ˘ C0 , δ)||AH ||2 · Tr((Aφ )2 )H −2C1 · C(n, ξ φ 2 s ˘ −4C1 · C(n, C0 , δ)H · Tr ((A ) )H ◦ AH ξ

˘ C0 , δ)Tr•g R(AH •, •) + 9b · (H s )3 · Tr((Aφ )2 )H . −2C1 · C(n, ξ H

Also, in similar to (9.5), we obtain 2 s |gH p(grad(||AH || ), grad H )| ≤ 2n C0 ||∇H AH ||2 · (H s )1−δ/2 .

This implies together with (9.7) that (9.9)

2

s

|gH (grad(||AH || ), grad H )| ≤

2(n − 1) s ˘ · H + C(n, C0 , δ) ||∇H AH ||2 . 3n

Denote by T 1 V the unit tangent bundle of V . Define a function Ψ over T 1 V by Ψ(X) := || d(Tr(AφX )2 )He ) ||

(X ∈ T 1 V ).

b 1 := sup max ||grad(Tr((Aφ )2 )H )||, which is It is clear that Ψ is continuous. Set K ξ t∈[0,T ) M

finite because Ψ is continuous and the closure of 54

∪

t∈[0,T )

φ(ft (M )) is compact. Also,

we have (9.10)

Tr•gH R

Hs Hs b AH − · id (•), • ≤ K2 · AH − · id n n

b 2 because of the homogeneity of N . By using (9.7), (9.9), (9.10), for some positive constant K 1 s 2 s ||AH || ≤ H , n ||gradH || ≤ ||∇H AH ||2 and Proposition 8.2, we can derive

(9.11)

∂ρ − △H ρ ∂t 4(n − 1) 2(n − 1)C1 − + 6nb H s · ||∇H AH ||2 ≤ 3n + 3n 3n H ˘ ˘ C0 , δ)(H s )4 +2C(n, C0 , δ) · ||∇ AH ||2 + 3C0 · C1 (H s )5−δ + 2C1 · C(n, b b 1 ||grad H s || + 3K1 · ||grad H s ||2 −3b(H s )5 + 6K Hs p s 3−δ b b 1 (H s )3−δ/2 +2C0 · C1 · K + 4C1 · C0 · K p1 (H ) s 2−δ/2 b 1 · (H s )3−δ b 2 · C0 · (H ) + 3C0 · C1 · K +2C1 · K ˘ C0 , δ) · K b 1 · (H s )2 + 4C1 · C(n, ˘ C0 , δ) · K b 1 · (H s )3 +2C1 · C(n, ˘ C0 , δ) · K b 2 · H s + 9b · K b 1 · (H s )3 . +2C1 · C(n,

Furthermore, by using the Young’s inequality (9.6) and the fact that C1 is sufficiently big compared to n and b, we can derive that ∂ρ b1 , K b2) − △H ρ ≤ C3 (n, C0 , C1 , b, δ, K ∂t

b1 , K b 2 ) only on n, C0 , C1 , b, δ, K b1 holds for some positive constant C3 (n, C0 , C1 , b, δ, K b 2 . This together with T < ∞ implies that and K b1 , K b 2 )t max ρt ≤ max ρ0 + C3 (n, C0 , C1 , b, δ, K M M b1 , K b2) · T ≤ max ρ0 + C3 (n, C0 , C1 , b, δ, K M

(0 ≤ t < T ). Therefore, we obtain

b1 , K b 2 ) · T · H s. ||grad H s ||2 ≤ b(H s )4 + max ρ0 · H s + C3 (n, C0 , C1 , b, δ, K M

Furthermore, by using the Young inequality (9.6), we obtain

b1 , K b2 , T ) ||grad H s ||2 ≤ 2b(H s )4 + C4 (n, C0 , C1 , b, δ, K

b1 , K b 2 , T ) only on n, C0 , C1 , b, δ, K b1 holds for some positive constant C4 (n, C0 , C1 , b, δ, K b b b K2 and T . Since b is any positive constant and C4 (n, C0 , C1 , b, δ, K1 , K2 , T ) essentially depends only on n and f0 , we obtain the statement of Proposition 9.1. 55

10

Proof of Theorem A.

In this section, we shall prove Theorem A. G. Huisken ([Hu2]) obtained the evolution inequality for the squared norm of all iterated covariant derivatives of the shape operators of the mean curvature flow in a complete Riemannian manifold satisfying curvature-pinching conditions in Theorem 1.1 of [Hu2]. See the proof of Lemma 7.2 (Page 478) of [Hu2] about this evolution inequality. In similar to this evolution inequality, we obtain the following evolution inequality. Lemma 10.1. For any positive integer m, the family {||(∇H )m AH ||2 }t∈[0,T ) satisfies the following evolution inequality:

(10.1)

∂||(∇H )m AH ||2 − △H ||(∇H )m AH ||2 ∂t H m ≤ −2||(∇ ) AH ||2 + C4 (n, m)  X × ||(∇H )i AH || · ||(∇H )j AH || · ||(∇H )k AH || · ||(∇H )m AH || i+j+k=m

+C5 (m)

X

i≤m



||(∇H )i AH || · ||(∇H )m AH || + C6 (m)||(∇H )m AH || ,

where C4 (n, m) is a positive constant depending only on n, m and Ci (m) (i = 5, 6) are positive constants depending only on m. In similar to Corollary 12.6 of [Ha], we can derive the following interpolation inequality. ∗ (T (1,1) M )) such that, for any t ∈ [0, T ), Lemma 10.2. Let S be an element of Γ(πM St is a G-invariant (1, 1)-tensor field on M . For any positive integer m, the following inequality holds: Z Z 2m/i v ≤ C(n, m) · max ||SH ||2(m/i−1) · ||(∇H )i SH )||B d¯ ||(∇H )m SH )||2B d¯ v, M

M

M

where C(n, m) is a positive constant depending only on n and m. From these lemmas, we can derive the following inequality.

56

Lemma 10.3. For any positive integer m, the following inequality holds: Z Z d H m 2 ||(∇ ) AH ||B d¯ v+2 ||(∇H )m+1 AH ||2B d¯ v dt M M

≤ C7 (n, m, C6 (m), Vol(M0 )) · max ||AH ||2 + 1 M Z 1/2 ! Z H m 2 H m 2 , ||(∇ ) AH ||B d¯ v+ ||(∇ ) AH ||B d¯ v ×

(10.2)

M

M

where C7 (n, m, C6 (m), Vol(M0 )) is a positive constant depending only on n, m, C6 (m) and the volume Vol(M0 ) of M0 = f0 (M ). Proof. By using (10.1) and the generalized Hölder inequality, we can derive Z Z d H m 2 ||(∇ ) AH ||B d¯ v+2 ||(∇H )m+1 AH ||2B d¯ v dt M M  i  2m j Z 2m X Z 2m 2m j H i H j i   v v ≤ C4 (n, m) · · ||(∇ ) AH ||B d¯ ||(∇ ) AH ||B d¯ ×

Z

i+j+k=m M

M

2m k

H k

k Z 2m · d¯ v

H m

AH ||2B

||(∇ ) AH ||B ||(∇ ) M i Z X Z 2m 2m H i i e +C(n, m)C(m) ||(∇ ) AH ||B d¯ v · M

i≤m

e +C(n, m)C(m + 1) ·

M

Z

H m

M

||(∇ )

AH ||2B

1 Z 2 d¯ v ·

M

M

1 2 d¯ v H m

2m 2m−i

||(∇ ) AH ||B 1 2 d¯ v .

2m−i 2m d¯ v

From this inequality and Lemma 10.2, we can derive the desired inequality. From this lemma, we can derive the following statement. Proposition 10.4. The family {||AH ||2 }t∈[0,T ) is not uniform bounded. Proof. Suppose that sup max ||AH ||2 < ∞. Denote by CA this supremum. Define t∈[0,T ) M

a function Φ over [0, T ) by Z Φ(t) :=

M

||(∇H )m (AH )t ||2B d¯ vt

(t ∈ [0, T )).

Then, according to (10.2), we have dΦ ≤ C7 (n, m, C6 (m), Vol(M0 )) · (CA + 1) · (Φ + Φ1/2 ). dt 57

Assume that supt∈[0,T ) Φ > 1. Set E := {t ∈ [0, T ) | Φ(t) > 1}. Take any t0 ∈ E. Then Φ ≥ 1 holds over [t0 , t0 + ε) for some a sufficiently small positive number ε. Hence we have dΦ ≤ 2C7 (n, m, C6 (m), Vol(M0 )) · (CA + 1) · Φ dt on [t0 , t0 + ε). From this inequality, we can derive Φ(t) ≤ Φ(t0 )e2C7 (n,m,C6 (m),Vol(M0 ))·(CA +1)(t−t0 )

(t ∈ [t0 , t0 + ε))

and hence Φ(t) ≤ Φ(t0 )e2C7 (n,m,C6 (m),Vol(M0 ))·(CA +1)T

(t ∈ [t0 , t0 + ε)).

This fact together with the arbitrariness of t0 implies that Φt is uniform bounded. Thus, we see that Z ||(∇H )m (AH )t ||2B d¯ vt < ∞ sup t∈[0,T )

M

holds in general. Furthermore, since this inequality holds for any positive integar m, it follows from Lemma 10.2 that Z ||(∇H )m (AH )t ||lB d¯ vt < ∞ sup t∈[0,T ) M

holds for any positive integar m and any positive constant l. Hence, by the Sobolev’s embedding theorem, we obtain sup max ||(∇H )m (AH )t || < ∞.

t∈[0,T ) M

Since this fact holds for any positive integer m, ft converges to a C ∞ -embedding fT as t → T in C ∞ -topology. This implies that the mean curvature flow ft extends after T because of the short time existence of the mean curvature flow starting from fT . This contradicts the definition of T . Therefore we obtain sup max ||AH ||2 = ∞.

t∈[0,T ) M

By imitating the proof of Theorem 4.1 of [A1,2], we can show the folowing fact, where we note that more general curvature flows (including mean curvature flows as special case) is treated in [A1,2]. 58

Lemma 10.5. The following uniform boundedness holds: inf max {ε > 0 | (AH )t ≥ εHts · id on M } > 0

t∈[0,T )

and hence

λmax (x, t) 1 ≤ , ε0 (x,t)∈M ×[0,T ) λmin (x, t) sup

where λmax (x, t) (resp. λmin (x, t)) denotes the maximum (resp. minimum) eigenvalue of (AH )(x,t) and ε0 denotes the above infimum. Proof. Since

∂hH H − △H hH (X, Y ) ∂t s

= −2H · hH (AH (X), Y ) + gH

∂AH H − △H AH (X), Y . ∂t

From this relation, Lemmas 5.5 and 5.8, we can derive ∂AH − △H H AH ∂t = −2H s ((Aφξ )2 )H + Tr (AH )2 − ((Aφ )2 )H · AH − R♯ .

Furthermore, from this evolution equation and Lemma 5.8, we can derive ∂ AH H AH − △H s ∂t H s H AH 1 ||grad H s ||3 · AH − 2((Aφξ )2 )H = s ∇H + grad H s s s 3 H H (H ) 2 1 φ 2 + s · Tr((Aξ ) )H · AH − s R♯ . H H For simplicity, we set SH := gH and P (S)H :=

1 AH (•), • Hs

||grad H s ||3 · hH − 2(((Aφξ )2 )H )♭ (H s )3 2 1 + s · Tr((Aφξ )2 )H · hH − s R, H H

where (((Aφξ )2 )H )♭ is defined by (((Aφξ )2 )H )♭ (•.•) := gH (((Aφξ )2 )H (•), •). Also, set ε0 := max{ε > 0 | (SH )0 ≥ εgH }. 59

Then, for any (x, t) ∈ M × [0, T ), any ε > 0 and any X ∈ Ker(SH + εgH )(x,t) , we can show P (SH + εgH )(x,t) (X, X) ≥ 0. Hence, by Theorem 6.1 (the maximum principle), we can derive that (SH )t ≥ ε0 gH , that is, (AH )t ≥ ε0 Hts gH holds for all t ∈ [0, T ). From this fact, it follows that λmin (x, t) ≥ ε0 ||H(x,t) || holds for all (x, t) ∈ M ×[0, T ). Hence we obtain λmax (x, t) λmax (x, t) 1 ≤ sup ≤ . ε0 (x,t)∈M ×[0,T ) λmin (x, t) (x,t)∈M ×[0,T ) ε0 ||H(x,t) || sup

According to this lemma, we see that such a case as in Figure 3 does not happen.

almost cylindrical part

t→T

Mt M0

Figure 8 : The case where lim M t is not a round point t→T

By using Proposition 10.4 and Lemma 10.5, we shall prove the statement (i) of Theorem A. Proof of (i) of Theorem A. According to Proposition 10.4 and Lemma 10.5, we have lim min λmin (x, t) = ∞.

t→T x∈M

Set Λmin (t) := minx∈M λmin (x, t). Let xmin (t) be a point of M with λmin (xmin (t), t) = Λmin (t) and set x ¯min (t) := φM (xmin (t)). Denote by γf¯t (¯xmin (t)) the normal geodesic xmin (t)). Set pt := γf¯t (¯xmin (t)) (1/Λmin (t)). Since N is of of f¯t (M ) starting from f¯t (¯ non-negative curvature, the focal radii of M t along any normal geodesic are smaller 1 ¯ than or equal to Λmin (t) . This implies that ft (M ) is included by the geodesic sphere 1 1 of radius Λmin (t) centered at pt in N . Hence, since lim Λmin (t) = 0, we see that, as t→T

t → T , M t collapses to a one-point set, that is, Mt collapses to a G-orbit. 60

Denote by (RicM )t the Ricci tensor of g t and let RicM be the element of ∗ (T (0,2) M )) defined by (Ric ) ’s. To show the statement (ii) of Theorem A, we Γ(πM M t prepare the following some lemmas. Lemma 10.6. (i) For the section RicM , the following relation holds: (10.3)

2

RicM (X, Y ) = −3Tr(AφX L ◦ AφY L )H − g(A X, Y ) + ||H|| · g(AX, Y )

∗ (T M ))), where where X L (resp. Y L ) is the horizontal lift of X (resp. (X, Y ∈ Γ(πM Y ) to V . (ii) Let λ1 be the smallest eigenvalue of A(x,t) . Then we have

(RicM )(x,t) (v, v) ≥ (n − 1)λ21 g (x,t) (v, v)

(10.4)

(v ∈ Tx M ).

Proof. Denote by Ric the Ricci tensor of N . By the Gauss equation, we have 2

RicM (X, Y ) = Ric(X, Y ) − g(A X, Y ) + ||H||g(AX, Y ) − R(ξ, X, Y, ξ) (X, Y ∈ T M). Also, by a simple calculation, we have Ric(X, Y ) = −3Tr(AφX L ◦ AφY L )H + 3gH ((AφX L ◦ AφY L )(ξ), ξ) and

R(ξ, X, Y, ξ) = 3gH ((AφX L ◦ AφY L )(ξ), ξ)

∗ (T M ))). From these relations, we obtain the relation (10.3). (X, Y ∈ Γ(πM

Next we show the inequality in the statement (ii). Since AφvL is skew-symmetric, we have Tr((AφvL )2 ) ≤ 0. Also we have 2

−g(x,t) (A(x,t) (v), v) + ||H (x,t) || · g (x,t) (A(x,t) (v), v) ≥ (n − 1)λ21 · g (x,t) (v, v). Hence, from the relation in (i), we can derive the inequality (10.4). By remarking the behavior of geodesic rays reaching the singular set of a compact Riemannian orbifold (see Figure 4) and using the discussion in the proof of Myers’s theorem ([M]), we can show the following Myers-type theorem for Riemannian orbifolds.

61

Theorem 10.7. Let (N, g) be an n-dimensional compact (connected) Riemannian orbifold. If its Ricci curvature Ric of (N, g) satisfies Ric ≥ (n − 1)K for some positive constant K, then the first conjugate radius along any geodesic in (N, g) is π smaller than or equal to √ and hence so is also the diameter of (N, g). K By using Propositions 9.1, 10.4, Lemmas 10.6 and Theorem 10.7, we prove the statement (ii) of Theorem A. γ1 (s2 )L γ2 (s′4 )L ei U

γ2 (s′3 )L

γ1 (s1 )L γ2 (s′1 )L γ2 (s′2 )L ϕ−1 i ◦ πΓi γ2 (s′2 ) γ2 (s′3 )

Ui γ2 (s′1 )

γ1 (s1 ) γ1 γ1 (s2 ) γ2 (s′4 )

γ2 γ1 (0) = γ2 (0) The case where g|Ui is a flat metric

Figure 9 : The behavior of geodesic rays around the singular set Proof of (ii) of Theorem A. (Step I) According to Proposition 9.1, for any positive

62

constant b, there exists a constant C(b, f0 ) (depending only on b and f0 ) satisfying ||grad H s ||2 ≤ b · (H s )4 + C(b, f0 ) on M × [0, T ). According to Proposition 10.4, we have limt→T (Hts )max = ∞. Hence there exists a C(b, f0 ) 1/4 s positive constant t(b) with (Ht )max ≥ for any t ∈ [t(b), T ). Then we b have √ (10.5) ||grad Hts || ≤ 2b(Hts )2max for any t ∈ [t(b), T ). Fix t0 ∈ [t(b), T ). Let xt0 be a maximal point of ||Ht0 ||. Take any geodesic γ of length √2||H ||1 ·b1/4 starting from xt0 . According to (10.5), we t0 max

have

||Ht0 || ≥ (1 − b1/4 )||Ht0 ||max along γ. From the arbitrariness of t0 , this fact holds for any t ∈ [t(b), T ). (StepII) For any x ∈ M , denote by γf (x) the normal geodesic of f t (M ) starting t

from f t (x). Set pt := γf (x) (1/λmin (x, t)) and qt (s) := γf (x) (s/λmax (x, t)). Since t t N is of non-negative curvature, the focal radii of f t (M ) at x are smaller than or equal to 1/λmin (x, t). Denote by G2 (T N ) the Grassmann bundle of N of 2-planes and Sec : G2 (T N ) → R the function defined by assigning the sectional curvature of Π to each element Π of G2 (T N ). Since ∪ f t (M ) is compact, there exists the t∈[0,T )

maximum of Sec over

∪

t∈[0,T )

f t (M ). Denote by κmax this maximum. It is easy to

c/λmax (x, t) show that the focal radii of f t (M ) at x are bigger than or equal to b for some positive constant b c depending only on κmax . Hence a sufficiently small neighborhood of f t (x) in f t (M ) is included by the closed domain surrounded by the geodesic spheres of radius 1/λmin (x, t) centered at pt and that of radius b c/λmax (x, t) centered at qt (b c). On the other hand, according to Lemma 10.5, we have λmax (x, t) < ∞. (x,t)∈M ×[0,T ) λmin (x, t) sup

By using these facts, we can show sup t∈[0,T )

(Hts )max 0 | (AH )t ≥ εHts · id on M } > 0.

t∈[0,T )

63

Set C0 := sup t∈[0,T )

(Hts )max (Hts )min

and ε0 := inf max {ε > 0 | (AH )t ≥ εHts · id on M }. t∈[0,T )

s · id on M × [0, T ), it follows from (ii) of Lemma 10.6 that Then, since AH ≥ ε0 Hmin

(RicM )(x,t) (v, v) ≥ (n − 1)ε20 · (Hts )2min · g (x,t) (v, v) for any (x, t) ∈ M × [0, T ) and any v ∈ Tx M . Hence, according to Theorem 10.7, the first conjugate radius along any geodesic γ in (M , gt ) is smaller equal to than or π π =M ε0 (Hts )min for any t ∈ [0, T ). This implies that expf t (x) Bf t (x) ε (H s ) 0 t min holds for any t ∈ [0, T ), where expft (x) denotes the exponential map of (M , g t ) at π denotes the closed ball of radius ε0 (Hπs )min in Tf (x) M f t (x) and Bf (x) t t t ε0 (Hts )min centered at the zero vector 0. By the arbitrariness of b (in (Step I)), we may assume ε4 that b ≤ 4π40C 4 . Then we have 0

√

1 2(Hts )max

· b1/4

≥

π ε0 (Hts )min

(t ∈ [0, T )). Let t0 be as in Step I. Then it follows from the above facts that ||Ht0 || ≥ (1 − b1/4 )||Ht0 ||max holds on M . From the arbitariness of t0 , it follows that s H s ≥ (1 − b1/4 )Hmax

holds on M × [t(b), T ). In particular, we obtain s Hmax 1 ≤ s Hmin 1 − b1/4

on M × [t(b), T ). Therefore, by approaching b to 0, we can derive lim

t→T

(Hts )max = 1. (Hts )min

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11

Proof of Theorem B

In this section, we shall prove Theorem B. Proof of Theorem B. Let M be the strictly convex hypersurface in G as in the statement of Theorem B. Set M := φ−1 (M ), where φ denotes the parallel transport map for G. Since the bi-invariant metric of the compact semi-simple Lie group G is of non-negative sectional curvature and locally symmetric, we can show L = 0, where L is as stated in Introduction. From the strictly covexity of M and L = 0, we can derive that M satisfies the condition (H s )2 hH > 2n2 LgH . Also, M (= φ(M )) satisfies the conditions (∗1 ) and (∗2 ) by the assumption. Therefore, by using Theorem A, 1 the regularized mean curvature flow starting from M collapses to a ΩH e (G)-orbit 0 in finite time. On the other hand, since M ⊂ holc (AH P ) by the assumption, we −1 −1 c have φ(µcϕ (hol−1 c (M ))) = L. Also, since holc (M ) is Gx0 -invariant, µϕ (holc (M )) is 1 −1 c −1 ΩH e (G)-invariant (see Section 2). Hence we obtain µϕ (holc (M )) = φ (M ). The statement of Theorem B follows from these facts.

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