WILLMORE BLOW-UPS ARE NEVER COMPACT ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A, Abstract. We prove a Lojasiewicz-Simon gradient inequality for the Willmore functional near Willmore surfaces and apply this inequality to exclude compact blow-ups for the Willmore flow.
1. Introduction For a smooth immersed closed surface f : Σ → Rn the Willmore functional is defined by Z 1 |H|2 dµf , W(f ) = 4 Σ where H denotes the mean curvature vector of f and µf is the measure induced by the pull-back metric g = f ∗ geuc . The Gauß equations and the Gauß-Bonnet Theorem give rise to equivalent expressions: Z Z 1 1 2 |A| dµf + 2π(1 − p(Σ)) = |A0 |2 dµf + 4π(1 − p(Σ)), W(f ) = 4 Σ 2 Σ where A denotes the second fundamental form, A0 = A − 12 g ⊗ H its tracefree part and p(Σ) is the genus of Σ. The Willmore functional is scale invariant and moreover invariant under the full M¨obius group of Rn . Critical points of W are called Willmore surfaces or, more precisely, Willmore immersions. We always have W(f ) ≥ 4π with equality only for round spheres; see [13], [14, Theorem 7.2.2] in codimension one, that is n = 3. On the other hand, if W(f ) < 8π, then f is an embedding by an inequality of Li and Yau [7]. By a Willmore flow of the surface Σ we mean a family (ft )t of immersions ft : Σ → Rn which solves the geometric evolution equation (1.1)
1 ∂t ft + (∆⊥ H + Q(A0 )H) = 0. 2
Date: February 1, 2007. 2000 Mathematics Subject Classification. 35K22, 35J70, 35P05, 53A05, 26D10. Key words and phrases. Willmore flow, Lojasiewicz-Simon inequality, blow-up, convergence. Eva Faˇsangov´ a was supported by the grant MSM 0021620839. Reiner Sch¨ atzle was supported by the DFG Forschergruppe 469. 1
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ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
The normal Laplacian ∆⊥ and the semilinear term Q(A0 )H are defined below. We point out that this evolution equation is a gradient system for the Willmore functional, [4]. It is known that (1.1) is locally well-posed and smoothing for initial values f0 ∈ C 4,α (Σ). However, the question of the maximal existence time T and the behaviour of the Willmore flow near the maximal existence time is in general still open. By [5, Theorem 5.2], the Willmore flow of a sphere in R3 with initial energy ≤ 8π exists globally and converges to a round sphere (see [3], [11] for earlier results of this type). On the other hand, a counterexample from [9] shows that the Willmore flow can develop a singularity near the maximal existence time, the initial energy in this counterexample being > 8π, see also [1]. In [3, Section 4] and [5, pp. 348-349] a blow-up procedure was set up showing that for each sequence tj % T there exists a subsequence denoted again tj and rj > 0, xj ∈ Rn such that fj := rj−1 (ftj +c0 rj4 − xj ) converges, for given c0 = c0 (n) > 0 and after appropriate reparametrization, ˆ → Rn , smoothly on compact subsets of Rn to a Willmore immersion fW : Σ ˆ 6= ∅ is a complete surface without boundary. Moreover, it is possible where Σ to select tj % T in such a way that fW is non-trivial in the sense AfW 6≡ 0, that is fW does not parametrize a union of planes. For T < ∞, we always have rj → 0 by [4, Theorem 1.2], and the limit of fj , after reparametrization, is called blow-up. For T = ∞, we may have rj → 0, rj = 1 or rj → ∞, after rescaling and passing to appropriate subsequences. In case rj → 0, we still call the limit a blow-up, whereas in case rj → ∞, we call the limit a blow-down. ˆ is compact. We know In this article, we are interested in the case when Σ ˆ is compact, then Σ and Σ ˆ by [3, Lemma 4.3], if one of the components of Σ ˆ is compact. are diffeomorphic; in particular, Σ The main result of our article excludes compact blow-ups and blow-downs. Theorem 1.1. For any Willmore flow (ft )t of a closed surface Σ, none of the components of blow-ups or blow-downs is compact. This main result will follow from a global existence and convergence result for Willmore flows starting near compact Willmore surfaces and having energy above these Willmore surfaces; near is here to be understood with respect to the norm in W 2,2 ∩ C 1 (Σ). This global existence and convergence result is in turn a consequence of parabolic regularity and the fact that the Willmore functional satisfies the Lojasiewicz-Simon gradient inequality near every Willmore immersion.
WILLMORE BLOW-UPS ARE NEVER COMPACT
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As a byproduct of the proof of the main result we obtain that Willmore flows starting near local minimizers of the Willmore functional exist globally and converge to a Willmore surface which is again a local minimizer of the Willmore functional. Theorem 1.2. Let Σ be a closed surface and let fW : Σ → Rn be a Willmore immersion which locally minimizes the Willmore functional in C k (k ≥ 2) in the sense that there exists δ > 0 such that W(f ) ≥ W(fW ) whenever kf − fW kC k ≤ δ. Then there exists ε > 0 such that for any immersion f0 : Σ → Rn satisfying kf0 − fW kW 2,2 ∩C 1 < ε the corresponding Willmore flow (ft )t with initial data f0 exists globally and converges smoothly, after reparametrization by appropriate diffeomorphisms ≈ Φt : Σ −→ Σ, to a Willmore immersion f∞ which also minimizes locally the Willmore functional in C k , i.e. ft ◦ Φt → f∞ as t → ∞. We point out that if fW is a local minimizer of the Willmore functional in C , then it is also a local minimizer in W 2,2 ∩ C 1 (see Remark 5.2 below). For round spheres in codimension one Theorem 1.2 has been proved by Simonett [11], provided that the W 2,2 ∩ C 1 -norm is replaced by the stronger h2+α norm. Theorem 1.2 applies in particular also to the Willmore surfaces which minimize globally the Willmore functional among those surfaces with fixed genus. k
The fact that we can start with initial values which are close to a Willmore surface in the W 2,2 ∩ C 1 topology follows from an existence, uniqueness and regularity result for Willmore flows which is proved in the Appendix. 2. Preliminaries Let Σ be a closed surface, i.e. a compact, smooth, two-dimensional manifold without boundary. Given an immersion f : Σ → Rn (n ≥ 3) we define the pull-back metric g = gf by hX, Y ig = hDf · X, Df · Y ieuc . The corresponding Riemannian connection ∇ is given by the representation Df · ∇X Y = P DX (DY f ), where P = Pf denotes the orthogonal projection of Rn onto the tangent space Tf (p) f (Σ) ⊂ Rn , and X and Y denote abstract tangent vector fields. In local
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
4
coordinates, the projection P is given by the formula P x = g ij hx, ∂i f i∂j f,
(2.1)
x ∈ Rn .
The second fundamental form A = Af is given by A(X, Y ) = DX (DY f ) − Df · ∇X Y
(2.2)
= P ⊥ DX (DY f ), with P ⊥ = I − P . The mean curvature vector H = Hf is by definition the trace of A, H = A(ei , ei ), the (ei ) being any orthonormal basis in the tangent space Tp Σ. The trace-free part A0 of the second fundamental form is given by 1 A0 (X, Y ) = A(X, Y ) − hX, Y ig H. 2 We will also consider the normal connection ∇⊥ associated with f . It acts on normal vector fields φ : Σ → Rn (i.e. φ = P ⊥ φ) and is given by the formula ⊥ ∇⊥ X φ = P DX φ.
In the following we write ∇⊥ φ for the linear map X 7→ ∇⊥ X φ which maps the abstract tangent space into the normal space, and we define the corresponding scalar product h∇⊥ φ, ∇⊥ ψi := h∇⊥ φ(ei ), ∇⊥ ψ(ei )i = hP ⊥ Dei φ, P ⊥ Dei ψi, where (ei ) is any orthonormal basis in Tp Σ. In local coordinates, this yields h∇⊥ φ, ∇⊥ ψi = g ij h∂i φ, ∂j ψi − g ij g kl h∂i φ, ∂k f i h∂l f, ∂j ψi.
(2.3)
For any k ≥ 0, we let W k,2 (Σ; Rn )⊥ := {φ ∈ W k,2 (Σ; Rn ) : φ = Pf⊥ φ} be the subspace of all vector fields in the Sobolev space W k,2 (Σ; Rn ) which are normal along f . The normal Laplace operator ∆⊥ = ∆⊥ gf is the closed linear 2 n ⊥ operator on L (Σ; R ) defined by D(∆⊥ ) := {φ ∈ W 1,2 (Σ; Rn )⊥ : ∃ψ ∈ L2 (Σ; Rn )⊥ ∀v ∈ W 1,2 (Σ; Rn )⊥ Z Z ⊥ ⊥ h∇ φ, ∇ vi dµf = − hψ, vi dµf }, Σ ⊥
∆ φ := ψ; see also [12].
Σ
WILLMORE BLOW-UPS ARE NEVER COMPACT
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k,2 For k ≥ 3, we denote by Wimm (Σ; Rn ) the set of all immersions in W k,2 (Σ; Rn ). k,2 This set is open in W k,2 (Σ; Rn ). The Willmore functional W : Wimm (Σ; Rn ) → R is defined by Z 1 W(f ) := |H|2 dµf . 4 Σ In [4, Section 2], it was proved that for normal vector fields φ ∈ W 4,2 (Σ; Rn ) (i.e. φ = Pf⊥ φ)
d W(f + tφ)|t=0 dtZ 1 = h∆⊥ H + Q(A0 )H, φi dµf 2 Σ Z =: hδW(f ), φi dµf .
W 0 (f ) · φ = (2.4)
Σ 0
Here, the operator Q(A ) is acting on vector fields φ : Σ → Rn by Q(A0 )φ = A0 (ei , ej )hA0 (ei , ej ), φieuc , where again (ei ) is any local orthonormal frame. Formula (2.4) justifies that (1.1) is a gradient flow. 3. Lojasiewicz-Simon gradient inequality In this section we prove that the Willmore functional, when considered on the open set of all immersions in W 4,2 (Σ; Rn ), satisfies the so-called LojasiewiczSimon gradient inequality near every Willmore immersion. The LojasiewiczSimon gradient inequality for real-valued functions goes back to the classical result by Lojasiewicz [8] for real analytic functions on RN and to its generalization to the infinite-dimensional case by L. Simon [10]. In the case of the Willmore functional, it is just the inequality (3.2) below. The exponent θ from the Lojasiewicz-Simon gradient inequality is called Lojasiewicz exponent. As indicated in the equality (2.4), we abbreviate (3.1)
1 δW(f ) := (∆⊥ H + Q(A0 )H). 2
Theorem 3.1 (Lojasiewicz-Simon gradient inequality). Let fW : Σ → Rn be a Willmore immersion. Then there exist constants θ ∈ (0, 21 ], c1 , c2 ≥ 0 and σ > 0 such that for every f ∈ W 4,2 (Σ; Rn ) with kf − fW kW 4,2 ≤ σ one has (3.2)
|W(f ) − W(fW )|1−θ ≤ c1 kW 0 (f )kL2 (Σ;Rn ) ≤ c2 kδW(f )kL2 (Σ;Rn ) .
In other words, the Willmore functional satisfies the Lojasiewicz-Simon gradient inequality near every Willmore immersion.
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ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
Note that Theorem 3.1 remains true if fW is not a Willmore immersion, i.e. if W 0 (fW ) 6= 0. However, in this case, the statement is trivial by simple continuity arguments. We describe the idea of the proof of Theorem 3.1; the proof itself will be carried out at the end of this section. We will first restrict the Willmore functional to all immersions in f ∈ W 4,2 (Σ; Rn ) for which f − fW is normal along fW , and we will show that this restriction satisfies the Lojasiewicz-Simon gradient inequality. In order to prove the inequality, by [2, Corollary 3.11], it suffices to prove that the restriction of the Willmore functional is an analytic function, that its first derivative is analytic from W 4,2 (Σ; Rn )⊥ into L2 (Σ; Rn )⊥ , and that the second variation is a Fredholm operator between these two spaces. These properties of the restriction of the Willmore functional will be essentially proved in two lemmas. Having the Lojasiewicz-Simon gradient inequality for the restriction of the Willmore functional, Theorem 3.1 will follow by noting that the Willmore functional does not depend on the parametrization of the surface, i.e. W(f ) = ≈ W(f ◦ Φ) for every immersion f and every diffeomorphism Φ : Σ −→ Σ. For the definition of analytic functions between two Banach spaces, we refer to [15]. In the following lemma, L(Rn ) is the space of all linear operators on Rn . Lemma 3.2. Let Σ be a closed surface, and let X, Y : Σ → T Σ be abstract tangent vector fields. The following functions are well-defined and analytic: 4,2 (i) The function Wimm (Σ; Rn ) → W 3,2 (Σ) given by f 7→ hX, Y ig . 4,2 (ii) The function Wimm (Σ; Rn ) → W 3,2 (Σ; L(Rn )) given by
f 7→ Pf . 4,2 (iii) The function Wimm (Σ; Rn ) → W 2,2 (Σ; Rn ) given by
f 7→ A(X, Y ). 4,2 (iv) The function Wimm (Σ; Rn ) → W 2,2 (Σ; Rn ) given by
f 7→ H. 4,2 (v) The function F : Wimm (Σ; Rn ) → W 2,2 (Σ; Rn ) given by
f 7→ F (f ) = Q(A0 )H. 4,2 (vi) The function Wimm (Σ; Rn ) → L2 (Σ; Rn ) given by
f 7→ ∆⊥ H.
WILLMORE BLOW-UPS ARE NEVER COMPACT
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4,2 (vii) The function Wimm (Σ; Rn ) → C(Σ) given by
f 7→ %f , where %f dµ = dµf and µ is a fixed measure on Σ associated with a Riemannian metric. Proof. (i) This is immediate from the definition of the metric g which depends on Df in a quadratic way, and from the fact that W 3,2 is an algebra for the pointwise multiplication. (ii) This follows from the representation (2.1) of the projection in local coordinates, from (i) and again from the fact that W 3,2 is an algebra for the pointwise multiplication. (iii) This follows from the definition of the second fundamental form A (second line in (2.2)) and (ii). (iv) follows from (iii) and the definition of the mean curvature vector as the trace of A. (v) It suffices to note that the function from (v) is a polynomial in g and A, and then to use (i) and (iii). In fact, by (i) and (iii), these two functions belong to W 2,2 which is (like W 3,2 ) an algebra for the pointwise multiplication. (vi) When writing the normal Laplace operator in local coordinates, by using its definition and the identity (2.3), we obtain p 1 ∆⊥ φ = P ⊥ p ∇i ( |g|g ij ∇j φ) − g ij g kl h∇i φ, ∇k f i∇j ∇l f . |g| By the properties (i) and (ii), all the coefficients in this representation are at least continuous and depend analytically on f . The claim follows from this representation and (iv). (vii) Let (Ωi , ϕi ) be an atlas of Σ, and let (αi ) be a partition of unity subordinate to this atlas. For every continuous u : Σ → R we have Z XZ p u dµf = (αi |g| u) ◦ ϕ−1 i dx. Σ
i
ϕi (Ωi )
From this identity we can deduce that any two Riemannian measures are mutually absolutely continuous with p corresponding continuous densities which are associated with the function |g|. The claim thus follows from (i). Lemma 3.3. For every smooth immersion f : Σ → Rn the operator 2 4,2 (∆⊥ (Σ; Rn )⊥ → L2 (Σ; Rn )⊥ f) : W
is Fredholm of index 0.
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
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Proof. The normal Laplace operator ∆⊥ is associated with the bilinear form a : W 1,2 ⊥ × W 1,2 ⊥ → R given by Z a(φ, ψ) = hP ⊥ Dφ, P ⊥ Dψi dµf , Σ
i.e. Z
⊥
Z
h∆ φ, ψi dµf = − Σ
h∇⊥ φ, ∇⊥ ψi dµf = −a(φ, ψ)
Σ
for all sufficiently regular φ and ψ; see also [12]. The form a is bounded, symmetric and L2 -elliptic in the sense that a(φ, φ) + ω kφk2L2 ≥ η kφk2W 1,2 for every φ ∈ W 1,2 ⊥ and some constants ω, η > 0. In fact, by (2.1), by the Cauchy-Schwarz inequality, and since h∇l f, φi = 0, for every φ ∈ W 1,2 ⊥ ,
= = = ≥ ≥
a(φ, φ) + ω kφk2L2 Z Z 2 kDφk dµf − hP Dφ, Dφi + ωkφk2L2 ZΣ ZΣ kDφk2 dµf − g ij g kl h∇i φ, ∇k f ih∇l f, ∇j φi dµf + ωkφk2L2 ZΣ ZΣ kDφk2 dµf + g ij g kl h∇i φ, ∇k f ih∇j ∇l f, φi dµf + ωkφk2L2 Σ Σ Z C (1 − Cε) kDφk2 dµf + (ω − )kφk2L2 4ε Σ 1 kφk2W 1,2 2
if ε > 0 is small and ω > 0 is large enough. By the Lax-Milgram lemma (or: by Riesz-Fr´echet), the operator −∆⊥ + ω : D(∆⊥ ) → L2 ⊥ is an isomorphism; in particular, by symmetry of the form a, ∆⊥ is self-adjoint. Since D(∆⊥ ) ,→ W 1,2 ⊥ embeds compactly into L2 ⊥ , we obtain that ∆⊥ : D(∆⊥ ) → L2 ⊥ is Fredholm of index 0. The same is true for the square (∆⊥ )2 : D((∆⊥ )2 ) → L2 ⊥ . It remains to show that D((∆⊥ )2 ) = W 4,2 ⊥ . For this we show first that for every φ ∈ W 2,2 ⊥ one has (3.3)
∆⊥ φ = ∆φ + b(Dφ, φ),
where ∆ = ∆gf = g ij ∇i ∇j is the Laplace operator and b is a linear local operator with smooth coefficients.
WILLMORE BLOW-UPS ARE NEVER COMPACT
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For every φ ∈ W 2,2 ⊥ and every ψ ∈ C ∞ (Σ; Rn ) we have Z Z ⊥ h∆ φ, ψi dµf = h∆⊥ φ, P ⊥ ψi dµf Σ Σ Z = − hP ⊥ Dφ, DP ⊥ ψi dµf ZΣ Z (3.4) = − hDφ, Dψi dµf + hP Dφ, DP ⊥ ψi dµf + Σ ZΣ + hDφ, DP ψi dµf Σ
By definition of the Laplace operator, Z Z Z ij hDφ, Dψi dµf = g h∇i φ, ∇j ψi dµf = − h∆φ, ψi dµf . Σ
Σ
Σ
Since h∇l f, P ⊥ ψi = 0, we calculate for the second term on the right-hand side of (3.4), Z Z ⊥ g ij g kl h∇i φ, ∇k f i h∇l f, ∇j P ⊥ ψi dµf hP Dφ, DP ψi dµf = Σ Σ Z = − g ij g kl h∇i φ, ∇k f i h∇j ∇l f, P ⊥ ψi dµf Z Σ = hb1 (Dφ), ψi dµf . Σ
Similarly, since hφ, ∇l f i = 0, we calculate for the third term on the right-hand side of (3.4), Z Z g ij h∇i φ, ∇j (g kl hψ, ∇k f i∇l f )i dµf hDφ, DP ψi dµf = Σ ZΣ = hb2 (Dφ), ψi dµf + Σ Z + g ij g kl h∇i φ, ∇l f i h∇j ψ, ∇k f i dµf Z Σ = hb2 (Dφ), ψi dµf − Σ Z − g ij g kl hφ, ∇i ∇l f i h∇j ψ, ∇k f i dµf Z Σ = hb3 (Dφ, φ), ψi dµf . Σ
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ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
Summing up, (3.4) becomes Z Z ⊥ h∆ φ, ψi dµf = h∆φ + b(Dφ, φ), ψi dµf , Σ
Σ
so that (3.3) follows. The operator b is linear and local and has smooth coefficients due to the assumption that f is smooth. If ∆⊥ φ ∈ L2 ⊥ (and φ ∈ W 1,2 ⊥ ), then b(Dφ, φ) ∈ L2 and therefore, by (3.3), ∆φ ∈ L2 . This implies φ ∈ W 2,2 ⊥ , i.e. D(∆⊥ ) = W 2,2 ⊥ . Similarly, if ∆⊥ φ ∈ W 2,2 ⊥ (and φ ∈ W 2,2 ⊥ ), then b(Dφ, φ) ∈ W 1,2 and therefore, by (3.3), ∆φ ∈ W 1,2 , i.e. φ ∈ W 3,2 ⊥ . But then we actually have b(Dφ, φ) ∈ W 2,2 and therefore ∆φ ∈ W 2,2 so that φ ∈ W 4,2 ⊥ . This proves the required regularity. Proof of Theorem 3.1. In the first step we prove that there exist constants θ ∈ (0, 12 ], C ≥ 0 and σ > 0 such that for every φ ∈ W 4,2 (Σ; Rn )⊥ with kφkW 4,2 ≤ σ one has (3.5)
|W(fW + φ) − W(fW )|1−θ ≤ C kW 0 (fW + φ)kL2 (Σ;Rn ) ,
i.e. we prove that the shifted Willmore functional W(fW + ·), when restricted to the space W 4,2 ⊥ of vector fields which are normal to fW , satisfies the Lojasiewicz-Simon gradient inequality near 0. By [2, Corollary 3.11], it suffices to show that W(fW + ·) is analytic in a neighbourhood of 0 in W 4,2 ⊥ , that W 0 is analytic with values in L2 (Σ; Rn )⊥ , and that W 00 (fW ) is a Fredholm operator of index 0 from W 4,2 (Σ; Rn )⊥ into L2 (Σ; Rn )⊥ . 4,2 We firstly note that the Willmore functional is analytic on Wimm (Σ; Rn ) by Lemma 3.2 (iv) and (vii). Secondly, we recall from (2.4) that the first variation acts on normal vector fields φ ∈ W 4,2 (Σ; Rn ) (normal to f ) by Z 1 0 (3.6) W (f ) · φ = h∆⊥ H + Q(A0 )H, φi dµf . + Q(A0 )φi dµf . 2 Σ Hence, by Lemma 3.2 (v), (vi) and (vii), the first variation is analytic when 4,2 considered as a function from Wimm (Σ; Rn ) into L2 (Σ; Rn ). 0 Note also that since W (f ) · φ = 0 for every tangent field φ ∈ W 4,2 (Σ; Rn ) (i.e. φ = P φ) and since ∆⊥ H + Q(A0 )H is a normal vector field, we obtain
W 0 (f ) · φ = W 0 (f ) · (P ⊥ φ + P φ) Z 1 = h∆⊥ H + Q(A0 )H, P ⊥ φi dµf 2 Σ Z 1 = h∆⊥ H + Q(A0 )H, φi dµf , 2 Σ i.e. the relation (3.6) actually holds for every vector field φ ∈ W 4,2 (Σ; Rn ).
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We need this in order to calculate the second variation of the Willmore functional. Let φ, ψ ∈ W 4,2 (Σ; Rn )⊥ (normal to fW ). Then d 0 W 00 (fW )(φ, ψ) = W (fW + tψ) · φ|t=0 dt Z 1d = h∆⊥ Hf +tψ + Q(A0fW +tψ )HfW +tψ , φi dµfW +tψ |t=0 . 2 dt Σ fW +tψ W We calculate, using [4, Lemma 2.2], Z d Hf +tψ , φi dµfW +tψ |t=0 = h∆⊥ dt Σ fW +tψ W Z d = h∆⊥ Hf +tψ , Pf⊥W +tψ φi dµfW +tψ |t=0 dt Σ fW +tψ W Z d ⊥ h∇⊥ Hf +tψ , ∇⊥ = − fW +tψ PfW +tψ φi dµfW +tψ |t=0 dt Σ fW +tψ W Z (3.7) = h(∆⊥ )2 ψ, φi dµfW + Σ Z 1 + h∆⊥ (Q(A0 )ψ + hH, ψiH , φi dµfW − 2 ZΣ ⊥ ⊥ − hA(·, ei )h∇⊥ ei ψ, Hi + ∇ei ψhA(·, ei ), Hi, ∇ φi dµfW − ZΣ − h(Df · (∇⊥ ψ)∗ + (∇⊥ ψ)∗ · Df )∆⊥ H, φi dµfW − ZΣ ⊥ − h∇⊥ H, A(·, ei )h∇⊥ ei ψ, φi + ∇ei ψhA(·, ei ), φii dµfW + Σ Z + h∇⊥ H, ∇⊥ P ⊥ φi hH, ψi dµfW , Σ
and
d dt Z
(3.8)
0
Z Σ
hQ(A0fW +tψ )HfW +tψ , φi dµfW +tψ |t=0 = Z
hF (fW )ψ, φi dµfW −
= Σ
hQ(A0 )H, φi hH, ψi dµfW ,
Σ
where F is the function from Lemma 3.2 (v). The main term in the two equalities above is the first term on the right-hand side of (3.7). This bilinear term corresponds to the operator (∆⊥ )2 which, by Lemma 3.3, is a Fredholm operator from W 4,2 ⊥ into L2 ⊥ . Consider the second term on the right-hand side of (3.7). Since Q(A0 ) maps W 2,2 into itself, and since the embedding W 4,2 ,→ W 2,2 is compact, the operator ψ 7→ ∆⊥ Q(A0 )ψ is compact from W 4,2 ⊥ into L2 ⊥ . All the other
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ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
terms on the right-hand side of (3.7) depend only on ψ, Dψ and D2 ψ and thus they correspond to a compact operator from W 4,2 ⊥ into L2 ⊥ . The same holds true for the second term on the right-hand side of (3.8). Finally, by Lemma 3.2 (v) and the compactness of the embedding W 2,2 ,→ L2 , the derivative F 0 (f ) is a compact operator from W 4,2 into L2 . Summing up, we see that the second derivative W 00 (fW ) is a compact perturbation of a Fredholm operator from W 4,2 ⊥ into L2 ⊥ , and hence it is a Fredholm operator. By [2, Corollary 3.11], we obtain the inequality (3.5). In the second step, we prove the claim. Let σ, C, θ be the constants from the Lojasiewicz-Simon inequality (3.5) proved in the first step. There exists σ 0 > 0 such that for every f ∈ W 4,2 (Σ; Rn ) with kf − fW kW 4,2 ≤ σ 0 there exist ≈ a diffeomorphism Φ : Σ −→ Σ and φ ∈ W 4,2 (Σ; Rn )⊥ satisfying kφkW 4,2 ≤ σ and f ◦ Φ = fW + φ. Then W(f ) = W(f ◦ Φ) = W(fW + φ) and kW 0 (f )kL2 = kW 0 (f ◦ Φ)kL2 = kW 0 (f + φ)kL2 . The claim follows now from (3.5) and the fact that the metrics gf are uniformly equivalent in a C 1 -neighbourhood of fW so that kW 0 (f )kL2 and kδW(f )kL2 are comparable. 4. Global existence In this section, we apply the Lojasiewicz-Simon inequality to obtain an appropriate global existence and convergence result for the Willmore flow. Lemma 4.1. Let fW : Σ → Rn be a Willmore immersion of a closed surface Σ, and let k ∈ N, δ > 0. Then there exists ε > 0 such that the following is true: suppose that (ft )t is a Willmore flow of Σ satisfying kf0 − fW kW 2,2 ∩C 1 < ε and (4.1)
W(ft ) ≥ W(fW ) whenever kft ◦ Φt − fW kC k ≤ δ, ≈
for some appropriate diffeomorphisms Φt : Σ −→ Σ. Then this Willmore flow exists globally, that is, T = ∞, and converges, after ≈ ˜ t : Σ −→ reparametrization by appropriate diffeomorphisms Φ Σ, smoothly to a Willmore immersion f∞ , that is, ˜ t → f∞ as t → ∞. ft ◦ Φ Moreover, W(f∞ ) = W(fW ) and kf∞ − fW kC k ≤ δ. Proof. We may assume k ≥ 4 and, by Proposition A.1 in the Appendix, without loss of generality kf0 − fW kC k,α < ε
WILLMORE BLOW-UPS ARE NEVER COMPACT
13
for some α > 0 and some ε > 0 small enough. There exists a diffeomorphism ≈ Φ0 : Σ −→ Σ and a vector field N0 : Σ → Rn which is normal along fW such that f0 ◦ Φ0 = fW + N0 =: f˜0 . Moreover, (4.2) kN0 kC k,α = kf˜0 − fW kC k,α ≤ C ε for some constant C independent of ε, if ε is small enough. Next, we consider the evolution equation (4.3) ∂ ⊥ f˜t + δW(f˜t ) = 0, t
where f˜t = fW + Nt for some smooth family of vector fields Nt which are ˜ normal along fW and where ∂t⊥ = Pf⊥ ˜t ∂t is the normal projection (along ft ) of the time derivative. Recall that the normal projection Pf⊥ ˜t = I − Pf˜t is given through (2.1). Choosing locally a smooth basis ν1 , . . . , νn−2 of Pf⊥W Rn , writing Nt = ϕr,t νr , and recalling (2.4), we obtain ⊥ 0 2 (∂t ϕr,t ) Pf⊥ ˜t νr = −∆ H − Q(A )H ⊥ 2 ˜ = −g ij Pf⊥ ˜t ∇i Pf˜t ∇j ∆g ft + B(·, Nt , DNt , D Nt ) ⊥ m 2 ˜ ˜ = −g ij g kl Pf⊥ ˜t ∇i Pf˜t ∇j (∂k ∂l ft − Γkl ∂m ft ) + B(·, Nt , DNt , D Nt ) 2 3 = −g ij g kl Pf⊥ ˜t ∂ijkl Nt + B(·, Nt , DNt , D Nt , D Nt ) 2 3 = − g ij g kl ∂ijkl ϕr,t Pf⊥ ˜t νr + B(·, Nt , DNt , D Nt , D Nt ),
where B is a smooth function depending on fW and changing from line to line. For kf˜t − fW kC 1 ≤ δ small enough, g ij g kl is uniformly elliptic, and there exists a constant C > 0 such that |Pf⊥ ˜t N | ≥ |N | − |PfW N − Pf˜t N |
(4.4)
≥ C |N |
for N normal along fW ,
⊥ so that Pf⊥ ˜ ν1 , . . . , Pf˜ νn−2 forms a basis of Nf˜t Σ. Therefore, t
(4.5)
t
1 ∂t ϕr,t + g ij g kl ϕr,t = Br (·, ϕt , Dϕt , D2 ϕt , D3 ϕt ), 2
and by (4.2), (4.6)
kϕr,0 kC k,α ≤ C ε
for r = 1, . . . , n − 2. For Cε < σ < δ small enough (σ as in the Lojasiewicz-Simon gradient inequality; see Theorem 3.1. Without loss of generality, σ < δ), equation (4.3)
14
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
has a solution (f˜t )t on a maximal existence interval [0, T˜[ satisfying in addition (this is part of our definition of the maximal existence time T˜) (4.7) kf˜t − fW kC k ≤ σ < δ for all t ∈ [0, T˜[. By parabolic Schauder estimates (see [10]), we get successively from (4.5) and (4.6), (4.8)
kϕkH k+α,(k+α)/4 ≤ C,
and in particular (4.9)
kf˜t − fW kC k,α ≤ C
for all t ∈ [0, T˜[.
The flow (f˜t )t and the Willmore flow (ft )t are connected in the following way. The flow (f˜t )t is a solution of (4.3) which we write as ∂t f˜t + δW(f˜t ) + df˜t · ξt = 0 for some smooth family of abstract tangential vector fields (ξt ). Solving the ordinary differential equation ∂t Φt = ξt ◦ Φt for t ∈ [0, T˜[ on Σ, Φ0 = idΣ , we get a smooth family of diffeomorphisms Φt on Σ. We calculate ∂t (f˜t ◦ Φt ) = (∂t f˜t ) ◦ Φt + (df˜t ) ◦ Φt · ∂t Φt = (∂t f˜t ) ◦ Φt + (df˜t ) ◦ Φt · (ξt ◦ Φt ) = −δW(f˜t ) ◦ Φt = −δW(f˜t ◦ Φt ). Thus, (f˜t ◦ Φt ◦ Φ−1 ) is a Willmore flow with initial data f˜0 ◦ Φ0 ◦ Φ−1 = f0 . As the solution of the Willmore flow is unique, we conclude (4.10) ft = f˜t ◦ Φt ◦ Φ−1 and T˜ ≤ T. In particular, it suffices to prove that the flow (f˜t )t is global and converges as t → ∞ to a Willmore immersion f∞ having the desired properties. We recall that the Willmore flow is a gradient flow for the Willmore functional. In particular, d W(ft ) = −k∂t ft k2L2 (Σ;dµf ) ≤ 0, t dt so that W(ft ) = W(f˜t ) is decreasing. From (4.1) and (4.7), we see that W(f˜t ) ≥ W(fW ) for every t ∈ [0, T˜[. If W(ft ) = W(fW ) for some t ∈ [0, T˜[, then W(fs ) = W(fW ) for every s ∈ [t, T˜[ and fs = ft , f˜s = f˜t are stationary for s ∈ [t, T˜[ (we remark that the Willmore flow is analytic in time so that the Willmore flow is then stationary
WILLMORE BLOW-UPS ARE NEVER COMPACT
15
for all s ≥ 0 by the uniqueness theorem for analytic functions). This implies ˜ ˜ T˜ = ∞ = T with (4.10), and fs ◦ Φ−1 s ◦ Φ = fs → f∞ = ft is a Willmore immersion satisfying W(f∞ ) = W(fW ) and k f∞ − fW kC k < δ by (4.7). In this case, there remains nothing to prove. So we can assume that W(ft ) > W(fW ) for all t ∈ [0, T˜[. Let θ ∈ (0, 12 ] be as in the Lojasiewicz-Simon inequality (Theorem 3.1). Then, by (4.7) and the Lojasiewicz-Simon gradient inequality, for every t ∈ [0, T˜[, d |W(ft ) − W(fW )|θ dt = θ |W(ft ) − W(fW )|θ−1 δW(f˜t ) · ∂t f˜t −
= θ |W(ft ) − W(fW )|θ−1 kδW(ft )kL2 (Σ;dµf˜ ) k∂t⊥ f˜t kL2 (Σ;dµf˜ ) t
t
θ k∂ ⊥ f˜t kL2 (Σ;dµf˜ ) . ≥ t C t As the metrics gf˜t are uniformly equivalent to gfW , the preceeding inequality and (4.4) imply d |W(ft ) − W(fW )|θ for every t ∈ [0, T˜[. dt Integrating in time yields for every t ∈ [0, T˜[, kf˜t − fW kL2 (Σ) ≤ kf˜0 − fW kL2 (Σ) + C |W(f˜0 ) − W(fW )|θ ≤ Ckf˜0 − fW kθ 2 . (4.11)
k∂t f˜t kL2 (Σ) ≤ −C
C (Σ)
Interpolating with (4.9) yields, by (4.2), σ 0 0 kf˜t − fW kC k (Σ) ≤ C kf˜t − fW kθC 2 (Σ) ≤ C εθ ≤ for t ∈ [0, T˜[, 2 for some 0 < θ0 < θ, if ε > 0 is small enough. As the solution (f˜t )t is maximal with respect to (4.7), we obtain T˜ = ∞, i.e. (f˜t )t exists globally and kf˜t − fW kC k ≤ σ for every t ∈ R+ . Moreover, by (4.11), and since W is decreasing along (f˜t )t , ∂t f˜t ∈ L1 (R+ ; L2 (Σ)). As a consequence, lim f˜t =: f∞ exists in L2 (Σ). t→∞
Interpolating again with (4.9), this implies that lim f˜t = f∞ in C k (Σ), t→∞
and by parabolic Schauder estimates the convergence is even smooth, that is, lim f˜t = f∞ in C l (Σ) for every l ∈ N. t→∞
16
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
Since the Willmore flow is a gradient flow, the immersion f∞ is a Willmore immersion. By (4.7), kf∞ − fW kC k ≤ σ, the Lojasiewicz-Simon gradient inequality (Theorem 3.1) applies to f∞ and yields |W(f∞ ) − W(fW )|1−θ ≤ c2 kδW(f∞ )kL2 = 0, hence W(f∞ ) = W(fW ).
5. Proof of the main results Proof of Theorem 1.1. Let [0, T [, 0 < T ≤ ∞, be the maximal existence interval of the Willmore flow (ft )t . We recall from the Introduction that blow-ups and blow-downs were obtained according to [3, Section 4] and [5] by selecting sequences tj % T, rj → 0 or ∞, xj ∈ Rn such that the immersions fj := rj−1 (ftj +c0 rj4 − xj ) converge smoothly, after appropriate reparametrization, on compact subsets ˆ → Rn , where Σ ˆ 6= ∅ is a complete of Rn to a Willmore immersion fW : Σ surface without boundary. ˆ were compact. We assume, by contradiction, that one of the components of Σ ˆ∼ Then by [3, Lemma 4.3], which holds true for any rj > 0, we get that Σ = Σ are ≈ diffeomorphic and compact and for appropriate diffeomorphisms Φj : Σ −→ Σ fj ◦ Φj → fW
smoothly on Σ.
Hence for j large enough kfj ◦ Φj − fW kC 2 (Σ) < ε with ε > 0 is as in Lemma 4.1. Putting fj,t := rj−1 (ftj +rj4 t − xj ) ◦ Φj , (fj,t )t is the Willmore flow with initial data fj ◦ Φj and the maximal existence interval [0, rj−4 (T − tj )[. Clearly, for every t ∈ [0, rj−4 (T − tj )[, W(fj,t ) ≥
lim
s→rj−4 (T −tj )
W(fj,s ) = lim W(fs ) = lim W(fi ) = W(fW ) i→∞
s→T
by invariance of the Willmore functional and the smooth convergence above. Then Lemma 4.1 implies that (fj,t )t exists globally. Hence, T = ∞ and the original flow (ft )t exists globally, too. Moreover, fj,t ◦ Φj,t → fj,∞ smoothly for t → ∞ ≈
and appropriate diffeomorphisms Φj,t : Σ −→ Σ. For fixed j, we can write ti + c0 ri4 = tj + rj4 si with si → ∞ and ti → T = ∞, and obtain rj−1 (fti +c0 ri4 − xj ) ◦ Φj,si = fj,si ◦ Φj,si → fj,∞ smoothly for i → ∞.
WILLMORE BLOW-UPS ARE NEVER COMPACT
17
In particular, the diameters converge: diam fti +c0 ri4 (Σ) → rj diam fj,∞ (Σ) ∈]0, ∞[ for i → ∞. On the other hand, as ri−1 (fti +c0 ri4 − xi ) ◦ Φi → fW smoothly for i → ∞, we obtain ri−1 diam fti +c0 ri4 (Σ) → diam fW (Σ) ∈]0, ∞[ for i → ∞. This contradicts to ri → 0 or ∞ as i → ∞, and therefore none of the compoˆ is compact. nents of Σ Remark 5.1. As pointed out in the introduction for T = ∞, we may have rj → 0, rj = 1 or rj → ∞, after rescaling and passing to appropriate subsequences. ˆ contains a compact component, we If in the case rj = 1 the limit surface Σ get as in the previous proof by [3, Lemma 4.3] that (ftj − xj ) ◦ Φj → fW
smoothly on Σ ≈
for some tj % T = ∞ and appropriate diffeomorphisms Φj : Σ −→ Σ. Then as above W(ft ) ≥ W(fW ) for all t ≥ 0 by invariance of the Willmore functional and the smooth convergence, and Lemma 4.1 implies that the Willmore flows (ft − xj )t converge, for j large enough, and after reparametrization by appro≈ priate diffeomorphisms Φj,t : Σ −→ Σ to Willmore immersions fj,∞ : Σ → Rn : (ft − xj ) ◦ Φj,t → fj,∞
for t → ∞.
Moreover, fj,∞ → fW smoothly as j → ∞. From this we conclude first that xj is bounded, and we may assume xj = 0. Next, we see that the whole flow (ft )t converges after reparametrization for t → ∞ to a Willmore immersion. Proof of Theorem 1.2. If fW is a local minimizer as in Theorem 1.2, condition (4.1) from Lemma 4.1 is clearly satisfied throughout the flow. Then Lemma 4.1 yields global existence and convergence as in the claim with W(f∞ ) = W(fW ) and kf∞ − fW kC k ≤ δ. In particular, f∞ is a Willmore immersion and minimizes the Willmore functional locally in C k . Remark 5.2. If fW is a local minimizer of the Willmore functional in C k (Σ), then it is also a local minimizer in W 2,2 ∩ C 1 (Σ). In fact, assume that γ > 0 is such that kf − fW kC k < γ ⇒ W(f ) ≥ W(fW ). Let τ > 0 be as in Proposition 6.1 below, and assume that kf0 − fW kW 2,2 ∩C 1 < τ . Let (ft )t be the Willmore flow with initial data f0 . Then W(f0 ) ≥ W(ft ) for all t in the maximal existence interval and kft ◦ Φ − fW kC k < γ for some t ≈ and some diffeomorphism Φ : Σ −→ Σ. Hence, W(f0 ) ≥ W(ft ) = W(ft ◦ Φ) ≥ W(fW ).
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
18
6. Appendix 6.1. Regularization. In this appendix, we prove a short time regularization result for the Willmore flow which enables us to assume closeness of our initial data only in W 2,2 ∩ C 1 . Proposition 6.1. For any smooth immersion f∗ : Σ → Rn of a closed surface Σ and any k ∈ N, γ > 0 , there exists a τ > 0 such that for any smooth immersion f0 : Σ → Rn with k f0 − f∗ k(W 2,2 ∩C 1 )(Σ) < τ the corresponding Willmore flow (ft )t with initial data f0 satisfies for some ≈ time 0 < t ≤ γ in the existence interval and some diffeomorphism Φ : Σ −→ Σ k ft ◦ Φ − f∗ kC k (Σ) < γ. Proof. Suppose that the assertion is not true for some γ > 0 . Then there exists a sequence fm → f∗ in (W 2,2 ∩ C 1 )(Σ) such that the corresponding Willmore flows (fm,t )t with initial data fm,0 = fm satisfy k fm,t ◦ Φ − f∗ kC k (Σ) ≥ γ
(6.1)
for all 0 < t ≤ γ in the existence interval of (fm,t )t and all diffeomorphisms ≈ Φ : Σ −→ Σ . We choose 0 < % ≤ 1 satisfying Z |Af∗ |2 dµf∗ ≤ ε/2 for any B2% ⊆ Rn f∗−1 (B2% )
for some ε > 0 chosen below. For m large enough, we have k fm − −1 f∗ kL∞ (Σ) < % which implies that fm (B% (x)) ⊆ f∗−1 (B2% (x)) for any x ∈ Rn , 2,2 1 and as fm → f∗ in (W ∩ C )(Σ) , we see gfm → gf∗ uniformly on Σ √ √ and |Afm |2gfm gfm → |Af∗ |2gf∗ gf∗ in L1 (Σ) . This yields for every m large enough, which we will assume from now on always, and every x ∈ Rn Z Z 2 |Afm |gfm dµfm ≤ |Afm |2gfm dµfm ≤ −1 fm (B% (x))
Z ≤
f∗−1 (B2% (x))
√ √
|Af∗ |2gf∗ dµf∗ + |Afm |2gfm gfm − |Af∗ |2gf∗ gf∗
L1 (Σ)
f∗−1 (B2% (x))
≤ ε.
WILLMORE BLOW-UPS ARE NEVER COMPACT
19
By [4, Theorem 1.2], for ε ≤ ε0 (n) small enough, the Willmore flows (fm,t )t exist at least on an interval [0, T ] with T = c0 (n)%4 ≤ c0 (n) ≤ 1 and moreover they satisfy Z |Afm,t |2 dµfm,t ≤ Cε for any B% ⊆ Rn and any t ∈ [0, T ]. (6.2) −1 fm,t (B% )
Then, by the interior estimate from [3, Theorem 3.5], we get for Cε < ε0 (n) small enough that (6.3)
k ∇⊥,k Afm,t kL∞ (Σ) ≤ Ck ε1/2 t−(k+1)/4
for t ∈]0, T ],
where ∇⊥,k denotes k−times applying the normal covariant derivative in the normal bundle along fm,t . We conclude 1 ⊥ 0 |∂t fm,t | = ∆fm,t Hfm,t + Q(Afm,t )Hfm,t ≤ Cε1/2 t−3/4 2 and Zt k ∂s fm,s kL∞ (Σ) ds ≤ Cε1/2 t1/4 . k fm,t − fm kL∞ (Σ) ≤ 0
Choosing tm ∈ (0, T ] with tm → 0 and 1/4 Cε1/2 tm >k fm − f∗ kL∞ (Σ)
for m large enough, we see k fm,t − f∗ kL∞ (Σ) ≤ Cε1/2 t1/4
for t ∈ [tm , T ].
Together with k Afm,t kL∞ (Σ) ≤ Cε1/2 t−1/4 by (6.3), we can write fm,t ◦Φm,t = f∗ + Nm,t =: f˜m,t with normal fields Nm,t along f∗ and k f˜m,t − f∗ kL∞ (Σ) ≤ Cε1/2 t1/4 , k f˜m,t − f∗ kC 1 (Σ) ≤ Cε1/4 < δ
(6.4)
for tm ≤ t ≤ T , if δ = δ(f∗ ) is appropriate and ε small enough. For δ = δ(f∗ ) small enough, we see for the smooth pull-back metric g∗ = f∗∗ geuc that 1 g∗ ≤ gf˜m,t ≤ 2g∗ for t ∈ [tm , T ]. 2 The reparametrized Willmore flow ∂t⊥ f˜m,t +δW(f˜m,t ) = 0 for tm ≤ t ≤ T still satisfies (6.2), hence (6.3). Choosing locally a smooth basis ν1 , . . . , νn−2 of Nf∗ Σ and writing Nm,t = ϕr,m,t νr , we see in local charts for k ≥ 0 ∂ f˜m,t = ∂ k f∗ + k
k X l=0
∂ l ϕr,m,t ∗ ∂ k−l νr ,
20
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
where, as in [3], ϕ ∗ ψ denotes any form depending on ϕ and ψ in a universal, bilinear way. ⊥ As Pf⊥ ˜m,t ν1 , . . . , Pf˜m,t νn−2 forms a basis of Nf˜m,t Σ , we get for δ = δ(f∗ ) small enough |∂ k ϕm,t | ≤ C|∂ k ϕr,m,t Pf⊥ ˜m,t νr | ≤ k−1 k ˜ ϕm,t | + . . . |ϕm,t | + 1) ≤ ≤ Cf∗ (|Pf⊥ ˜m,t ∂ fm,t | + |∂ k ˜ k−1 ˜ fm,t | + . . . |f˜m,t | + 1). ≤ Cf∗ (|Pf⊥ ˜m,t ∂ fm,t | + |∂
˜ Recalling ∂ij f˜m,t = Af˜m,t ,ij + Γlf˜ ,ij ∂l f˜m,t in local charts and Pf⊥ ˜m,t ∂ fm,t = 0, m,t we see k−1 X k+1−l ˜ k k+2 ˜ ⊥ ⊥ fm,t , ∂ l Γ ∗ Pf⊥ Pf˜m,t ∂ fm,t = Pf˜m,t ∂ Af˜m,t + ˜m,t ∂ l=0 2
hence, as Γ = Γ(∂f, ∂ f ) , |∂ k+2 f˜m,t | ≤ Cf∗ (|∂ k+2 ϕm,t | + . . . |ϕr,m,t | + 1) ≤ k+2 ˜ ≤ Cf∗ (|Pf⊥ fm,t | + |∂ k+1 f˜m,t | + . . . |f˜m,t | + 1) ≤ ˜m,t ∂ k ≤ Cf∗ ,Λk+1 (|Pf⊥ ˜m,t ∂ Af˜m,t | + 1),
(6.5)
where |∂ k+1 f˜m,t | + . . . + |f˜m,t | ≤ Λk+1 . In particular for k = 0, by (6.3) and (6.4), (6.6) |∂ 2 f˜m,t |, |Γ| ≤ Cf∗ (1 + ε1/2 t−1/4 ) in local charts. Now we proceed as in the last step of the proof of [4, Theorem 1.2] and obtain inductively by (6.3) using (6.5) |∂ l ∇⊥,k−l A ˜ |, |∂ k+2 f˜m,t | ≤ Cf∗ ,k,τ for tm ≤ τ < 2τ ≤ t ≤ T. fm,t
From the parabolic evolution equation (4.3) for ϕm , we obtain as in (4.8) with interior estimates in time k f˜m,t kH k+α,(k+α)/4 (Σ×[2τ,T ]) ≤ Cf∗ ,k,τ . Recalling tm → 0 we see that (f˜m,t )t → (f˜t )t smoothly on compact subsets of ]0, T ] and f˜ is a smooth Willmore flow solving ∂t⊥ f˜t + δW(f˜t ) = 0 on ]0, T ] and satisfying R |Af˜t |2 dµf˜t ≤ Cε for any B% ⊆ Rn , B%
k f˜t − f∗ kL∞ (Σ) ≤ Cε1/2 t1/4 , k f˜t − f∗ kC 1 (Σ) ≤ δ, for all t ∈]0, T ] .
WILLMORE BLOW-UPS ARE NEVER COMPACT
21
Now we consider the Willmore flow ft = f∗ + Nt with Nt normal along f∗ and ∂t⊥ ft + δW(ft ) = 0 with initial data f0 := f∗ which exists on a time interval [0, T0 [ with 0 < T0 ≤ ∞ . As f˜t , ft → f∗ in L∞ (Σ) when t → 0 , we obtain from the uniqueness proposition below (Proposition 6.2) that f˜t = ft for 0 < t < min(T, T0 ) . As (ft )t is smooth, there exists 0 < t < min(T, T0 , γ) such that k ft − f∗ kC k (Σ) < γ. As fm,t ◦ Φm,t = f˜m,t → f˜t = ft smoothly, we see for m large enough k fm,t ◦ Φm,t − f∗ kC k (Σ) < γ which contradicts (6.1), and the proposition follows.
6.2. Uniqueness. It remains to state and proof the uniqueness proposition used in the above proof. Proposition 6.2. Let f∗ : Σ → Rn be a smooth immersion of a closed surface Σ and fm = f∗ + Nm : Σ×]0, T ] → Rn , m = 1, 2, be two Willmore flows with Nm normal along f∗ and ∂t⊥ fm,t + δW(fm,t ) = 0 satisfying k fm,t − f∗ kC 1 (Σ) ≤ ε, Z
|Afm,t |2 dµfm,t ≤ ε
for any B% ⊆ Rn ,
B%
for all t ∈]0, T ] and some % > 0 . If ε < ε0 (n, f∗ ) is small enough, then (6.7)
1/2
k f2,t − f1,t kL2 (Σ) ≤ Cf∗ tCf∗ ε
1/2
τ −Cf∗ ε
k f2,τ − f1,τ kL2 (Σ)
for 0 < τ ≤ t ≤ min(c(n)%4 , T ) . If, moreover, f2,t − f1,t → 0 weakly in L2 (Σ) for t → 0 then the flows coincide: f2,t = f1,t
for t ∈]0, T ].
Proof. We may assume that T ≤ c0 %4 for some c0 < c(n) small enough and after rescaling that % = 1 . We write gm , µm , Am , . . . for the metric, measure, second fundamental form, ... induced by fm for m = 1, 2 and for metric terms induced by g∗ := f∗∗ geuc for m = ∗ . The normal projections along f = fm , f∗ are denoted by Pf⊥ . By the interior estimate from [3, Theorem 3.5], we get for ε < ε0 (n), c0 < c(n) small enough that (6.8)
k ∇m,⊥,k Am,t kL∞ (Σ) ≤ Ck ε1/2 t−(k+1)/4 ,
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
22
where ∇m,⊥,k denotes k−times applying the normal covariant derivative with respect to gfm in the normal bundle along fm . We conclude for ε < ε(f∗ ) small enough that 1 gf ≤ g∗ ≤ 2gfm,t , 2 m,t
(6.9) and |∂t fm,t | ≤
2|∂t⊥ fm,t |
⊥ 0 = ∆gm,t Hm,t + Q(Am,t )Hm,t ≤ Cε1/2 t−3/4 .
In particular f2,t − f1,t converges strongly in L∞ (Σ) for t → 0 , and if f2,t − f1,t → 0 weakly in L2 (Σ) for t → 0 , then Zt (6.10)
k f2,t − f1,t kL∞ (Σ) ≤
k ∂s (f2,s − f1,s ) kL∞ (Σ) ds ≤ Cε1/2 t1/4 .
0
As in (6.6) we get from (6.8) (6.11)
|∂ 2 fm,t |, |∂gm,t |, |Γm − Γ∗ | ≤ Cf∗ (1 + ε1/2 t−1/4 )
in local charts. Since the difference of two Christoffel symbols is a tensor, the second estimate is invariant not depending on the chart or the metrics gfm or g∗ by (6.9). Putting h := f2 − f1 , we conclude (6.12) |Γ2 − Γ1 | ≤ C(|∂ 2 h| + |∇h| |∂ 2 f |) ≤ C|∇∗,2 h| + Cf∗ (1 + ε1/2 t−1/4 ) |∇h|, where the estimate of the term on the left by the term on the right is independent of the chart, as it involves only tensors. Next A2,ij − A1,ij = ∇2i ∇2j f2 − ∇1i ∇1j f1 = ∗,k 1,k ∗,k = ∇∗i ∇∗j h − (Γ2,k ij − Γij )∇k f2 + (Γij − Γij )∇k f1 =
= ∇∗i ∇∗j h + (Γ2 − Γ∗ ) ∗ ∇h + (Γ2 − Γ1 ) ∗ ∇f1 , hence (6.13)
|A2 − A1 | ≤ C|∇∗,2 h| + Cf∗ (1 + ε1/2 t−1/4 ) |∇h|.
⊥ Multiplying the evolution equations ∂t fm,t + δW(fm,t ) = 0 by ht = f2,t − f1,t and integrating with respect to µm , we get Z Z Z ⊥ ⊥ 2h∂t fm , hi dµm + h∆m Hm , hi dµm + hQ(A0m )Hm , hi dµm = Σ
(6.14)
Σ
Σ
=: I1m + I2m + I3m = 0 for m = 1, 2.
WILLMORE BLOW-UPS ARE NEVER COMPACT
23
For the first term, we calculate Z Z ⊥ 2 1 I1 − I1 = 2h∂t f2 , hi dµ2 − 2h∂t⊥ f1 , hi dµ1 = Σ
Σ
Z =
Z ⊥ ⊥ 2 h∂t f2 , Pf2 hi − h∂t f1 , Pf1 hi dµ1 + 2h∂t⊥ f2 , hi dµ2 − dµ1 = Σ
Σ
Z =
Z Z ⊥ ⊥ ⊥ ⊥ 2h∂t h, Pf1 hi dµ1 − 2h∂t f2 , (Pf2 −Pf1 )hi dµ1 + 2h∂t f2 , hi dµ2 − dµ1 ≥ Σ
Σ
Σ
Z ≥
(6.15)
2h∂t h, Pf⊥1 hi dµ1 − Cf∗ ε1/2 t−3/4
Z |∇h| |h| dµ∗ . Σ
Σ
By [4, Lemma 2.2 (2.16)] and a change of variable ∂t dµ = − hH, ∂t⊥ f i + divg ∂ttan f dµ, where ∂ttan f = ∂t f − ∂t⊥ f is the tangential part of ∂t f along f . We proceed Z d |Pf⊥ h|2 dµ = dt Σ
Z =
2h∂t⊥ Pf⊥ h, Pf⊥ hi dµ +
Σ
Z =
|Pf⊥ h|2 − hH, ∂t⊥ f i + divg ∂ttan f dµ.
Σ
2h∂t⊥ Pf⊥ h, Pf⊥ hi
Σ
Z
Z dµ −
|Pf⊥ h|2 hH, ∂t⊥ f i
Z dµ −
Σ
∇(|Pf⊥ h|2 )∂ttan f dµ ≤
Σ
(6.16) Z Z Z ⊥ ⊥ ⊥ 1/2 −3/4 −1 ≤ 2h∂t Pf h, Pf hi dµ+Cf∗ ε t |∇h| |h| dµ∗ +Cf∗ (1+εt ) |h|2 dµ∗ . Σ
Σ
Σ
We observe ∂t⊥ Pf⊥ h = ∂t⊥ h − g ij hh, ∂i f i∂t⊥ ∂j f and ⊥ ⊥ kl ⊥ kl ∂t⊥ ∂j f = ∇⊥ ∂ f = ∇ ∂ f +g h∂ f, ∂ f i∂ f = ∇⊥ t k l j t j t j ∂t f +g h∂t f, ∂k f iAjl = 1 ⊥ 0 = − ∇⊥ ∆ H + Q(A )H + g kl h∂t f, ∂k f iAjl , 2 j hence by (6.8) |∂t⊥ ∂j f | ≤ Cε1/2 t−1 .
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
24
Therefore Z Z Z ⊥ ⊥ 1/2 −1 ⊥ ⊥ ⊥ |h|2 dµ. 2h∂t Pf h, Pf hi dµ ≤ 2h∂t h, Pf hi dµ + Cε t Σ
Σ
Σ
Plugging into (6.16), we obtain Z d |Pf⊥ h|2 dµ ≤ dt Σ Z Z Z 1/2 −1 ⊥ ⊥ 1/2 −3/4 |∇h| |h| dµ∗ +Cf∗ (1+ε t ) |h|2 dµ∗ ≤ 2h∂t h, Pf hi dµ+Cf∗ ε t Σ
Σ
Σ
and with (6.15) (6.17) Z Z Z d 1 1/2 −1 2 1/2 −3/4 ⊥ 2 |∇h| |h| dµ∗ −Cf∗ (1+ε t ) |h|2 dµ∗ . I1 −I1 ≥ |Pf1 h| dµ1 −Cf∗ ε t dt Σ
Σ
Σ
By the equation of Mainardi-Codazzi and ∇j ∇i f = Aij , we see ⊥ ⊥ ⊥ ∇⊥ k ∇j ∇i f = ∇k Aij = ∇j Aki = ∇j ∇k ∇i f
and Z I2 =
ij kl
g g
⊥ h∇⊥ l ∇j ∇k ∇i f, hi
Z
⊥ ⊥ g ij g kl h∇k ∇i f, ∇⊥ j ∇l P hi dµ.
dµ =
Σ
Σ
We calculate ⊥ ⊥ ⊥ ⊥ rs ⊥ ⊥ rs ∇⊥ ∇ P h = ∇ ∇ h − g hh, ∂ f i∂ f = ∇ ∇ h − g hh, ∂ f iA r s r ls = j l j l j l rs ⊥ rs = ∇⊥ ∇ h − g h∇ h, ∂ f i∂ f − ∇ g hh, ∂ f iA l l r s r ls = j j = P ⊥ ∇j ∇l h−g rs h∇l h, ∂r f iAjs −g rs h∇j h, ∂r f iAls −g rs hh, Ajr iAls −g rs hh, ∂r f i∇⊥ j Als . ⊥ Again using the equation of Mainardi-Codazzi ∇⊥ j Als = ∇s Ajl , we get Z Z ij kl ⊥ I2 = g g h∇k ∇i f, P ∇j ∇l hi dµ − 2 g ij g kl g rs hAik , Ajs ih∂r f, ∇l hi dµ− Σ
Z −
Σ
g ij g kl g rs hAik , Als ihAjr , hi dµ −
Σ
Z =
g ij g kl h∇k ∇i f, ∇l ∇j hi dµ − 2
Σ
Z − Σ
1 g ij g kl g rs hAik , Als ihAjr , hi dµ+ 2
Z
ZΣ
ZΣ Σ
g ij g kl g rs hAik , ∇⊥ s Ajl ih∂r f, hi dµ = g ij g kl g rs hAik , Ajs ih∇r f, ∇l hi dµ− 1 |A| hH, hi dµ+ 2 2
Z Σ
g rs |A|2 h∇r f, ∇s hi dµ =
WILLMORE BLOW-UPS ARE NEVER COMPACT
Z =
g ij g kl h∇k ∇i f, ∇l ∇j hi dµ +
Z
25
A ∗ A ∗ ∇f ∗ ∇h + A ∗ A ∗ A ∗ h dµ.
Σ
Σ
Recalling the cubic form of Q(A0 )H in the second fundamental form and again that ∇k ∇i f = Aki is normal along f , we obtain (6.18) Z Z ij kl A ∗ A ∗ ∇f ∗ ∇h + A ∗ A ∗ A ∗ h dµ. I2 + I3 = g g h∇k ∇i f, ∇l ∇j hi dµ + Σ
Σ
Calling these integrals J2 and J3 , we proceed using (6.8), (6.11) - (6.13) Z Z ij kl 2 2 2 2 2 1 J2 − J2 = g2 g2 h∇k ∇i f2 , ∇l ∇j hi dµ2 − g1ij g1kl h∇1k ∇1i f1 , ∇1l ∇1j hi dµ1 = Σ
Σ
Z ≥
∗,s 2,s ∗,r ∗ ∗ g1ij g1kl h∇∗k ∇∗i f2 − (Γ2,r ki − Γki )∇r f2 , ∇l ∇j h − (Γlj − Γlj )∇s hi dµ1 −
Σ
Z −
∗,r 1,s ∗,s ∗ ∗ g1ij g1kl h∇∗k ∇∗i f1 − (Γ1,r ki − Γki )∇r f1 , ∇l ∇j h − (Γlj − Γlj )∇s hi dµ1 −
Σ
Z −C
|∇h| |A2 | |∇∗,2 h + (Γ2 − Γ∗ ) ∗ ∇h| dµ∗ ≥
Σ
Z ≥ c0,f∗
∗,2
Z
2
|∇ h| dµ∗ − C Σ
|hA2 , (Γ2 − Γ∗ ) ∗ ∇hi − hA1 , (Γ1 − Γ∗ ) ∗ ∇hi| dµ∗ −
Σ
Z D E 2 ∗ 1 ∗ ∗,2 −C (Γ − Γ ) ∗ ∇f2 − (Γ − Γ ) ∗ ∇f1 , ∇ h dµ∗ − Σ
−Cε
1/2 −1/4
Z
∗,2
|∇ h| |∇h| dµ∗ − Cε
t
1/2 −1/4
t
∗
2
k Γ − Γ kL∞ (Σ)
Σ
≥ c0,f∗
|∇∗,2 h|2 dµ∗ − Cf∗ (1 + εt−1/2 )
Σ
−C
|∇h|2 dµ∗ ≥
Σ
Z
Z
Z
Z
|∇h|2 dµ∗ −
Σ
|A2 − A1 | |Γ2 − Γ∗ | |∇h| dµ∗ − C
Σ
Z
|A1 | |Γ2 − Γ1 | |∇h| dµ∗ −
Σ
Z −C
|Γ2 − Γ∗ | |∇h| |∇∗,2 h| dµ∗ −
Σ
Z −C
2
1
∗,2
Z
|Γ − Γ | |h∇f1 − ∇f∗ , ∇ hi| dµ∗ − C Σ
Σ
|Γ2 − Γ1 | |h∇f∗ , ∇∗,2 hi| dµ∗ ≥
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
26
Z
∗,2
−1/2
2
|∇ h| dµ∗ − Cf∗ (1 + εt
≥ c0,f∗
Z )
|∇h|2 dµ∗ −
Σ
Σ
−Cf∗ (1 + Cε1/2 t−1/4 )
Z
C|∇∗,2 h| + Cf∗ (1 + ε1/2 t−1/4 ) |∇h| |∇h| dµ∗ −
Σ
−C k ∇f1 − ∇f∗ kL∞ (Σ)
Z
C|∇∗,2 h| + Cf∗ (1 + ε1/2 t−1/4 ) |∇h| |∇∗,2 h| dµ∗ −
Σ
−
Z
∗,2
C|∇ h| + Cf∗ (1 + ε
1/2 −1/4
t
) |∇h| |h∇f∗ , ∇∗,2 hi| dµ∗ ≥
Σ
(6.19) Z Z Z 2 ∗,2 2 −1/2 ) |∇h| dµ∗ −Cf∗ |h∇f∗ , ∇∗,2 hi|2 dµ∗ , ≥ c0,f∗ |∇ h| dµ∗ −Cf∗ (1+εt Σ
Σ
Σ
as k ∇f1 − ∇f∗ kL∞ (Σ) ≤ ε . For the last term, we recall that h = f2 − f1 = N2 − N1 ⊥ Nf∗ Σ , hence hh, ∇f∗ i = 0 . By differentiation, we get h∇i h, ∇k f∗ i + hh, A∗,ik i = 0, h∇∗j ∇∗i h, ∇k f∗ i + h∇i h, A∗,jk i + h∇j h, A∗,ik i + hh, ∇i∗,⊥ A∗,ik i = 0 and by smoothness of f∗ |h∇f∗ , ∇∗,2 hi| ≤ Cf∗ (|h| + |∇h|). Plugging into (6.19) yields (6.20) Z Z Z −1/2 ∗,2 2 2 2 1 ) |∇h| dµ∗ − Cf∗ |h|2 dµ∗ . J2 − J2 ≥ c0,f∗ |∇ h| dµ∗ − Cf∗ (1 + εt Σ
Σ
Σ
Next using (6.8), (6.13), we get for any δ > 0 |J32 − J31 | ≤ Z
Z
≤C ZΣ +C
|A2 − A1 | |A| |∇h| dµ∗ + C
|A|2 |∇h|2 dµ∗ +
Σ
|A2 − A1 | |A|2 |h| dµ∗ + C
Σ
|A|3 |∇h| |h| dµ∗ ≤
Σ
≤ Cε1/2 t−1/4 +Cεt−1/2
Z
Z
ZΣ Σ
C|∇∗,2 h| + Cf∗ (1 + ε1/2 t−1/4 ) |∇h| |∇h| dµ∗ +
C|∇∗,2 h| + Cf∗ (1 + ε1/2 t−1/4 ) |∇h| |h| dµ∗ +
WILLMORE BLOW-UPS ARE NEVER COMPACT
Z
+Cεt−1/2
|∇h|2 dµ∗ + Cε2 t−1
∗,2
−1/2
2
|∇ h| dµ∗ + Cf∗ ,δ (1 + εt
≤δ
27
|h|2 dµ∗ ≤
Σ
Σ
(6.21) Z
Z
Z
2 −1
2
Z
|∇h| dµ∗ + Cδ ε t
)
Σ
Σ
Σ
|h|2 dµ∗ .
Combining (6.14), (6.17), (6.18), (6.20), (6.21), we obtain Z Z d ⊥ 2 |Pf1 h| dµ1 + c0,f∗ |∇∗,2 h|2 dµ∗ ≤ dt Σ
Σ
≤ Cf∗ (1 + ε
1/2 −1/2
t
Z
2
|∇h| dµ∗ + Cf∗ (1 + ε
)
1/2 −1
= Cf∗ (1 + ε
t
Σ
Z h−∆∗ h, hi dµ∗ + Cf∗ (1 + ε
)
1/2 −1
≤δ
∗,2
Z
t )
Σ
Z
|h|2 dµ∗ =
t )
Σ 1/2 −1/2
Z
|h|2 dµ∗ ≤
Σ 2
|∇ h| dµ∗ + Cf∗ ,δ (1 + ε
1/2 −1
Z
|h|2 dµ∗ ,
t )
Σ
Σ
and d dt
Z
|Pf⊥1 h|2
Σ
Z
∗,2
2
|∇ h| dµ∗ ≤ Cf∗ (1 + ε
dµ1 + c0,f∗
1/2 −1
Σ
Z
t )
|h|2 dµ1 .
Σ
As |Pf⊥1 h| ≥ C|h| for ε < ε(f∗ ) small enough, integration from 0 < τ < t ≤ T yields Z
Zt
2
|ht | dµ1 ≤ C exp
Cf∗ (1 + ε
1/2 −1
s ) ds
τ
Σ
|hτ |2 dµ1 ≤
Σ
Cf∗ ε1/2 −Cf∗ ε1/2
≤ Cf∗ t
Z
Z
|hτ |2 dµ1
τ
Σ
and (6.7) follows. If f2,t − f1,t → 0 weakly in L2 (Σ) , we get from (6.10) 1/2
k f2,t − f1,t k2L2 (Σ,g∗ ) ≤ Cf∗ tCf∗ ε
1/2
τ −Cf∗ ε
ετ 1/2 → 0 for τ → 0,
when C(f∗ )ε1/2 < 1/2 , and therefore f2 − f1 = 0 .
28
ˇ ´ AND REINER SCHATZLE ¨ RALPH CHILL, EVA FASANGOV A,
References 1. S. Blatt, Beispiel eines blowup-limes f¨ ur den Willmorefluß, diploma thesis, Universit¨at Bonn. 2. R. Chill, On the Lojasiewicz-Simon gradient inequality, J. Funct. Anal. 201 (2003), 572–601. 3. E. Kuwert and R. Sch¨ atzle, The Willmore flow with small initial energy, J. Differential Geom. 57 (2001), 409–441. 4. E. Kuwert and R. Sch¨ atzle, Gradient flow for the Willmore functional, Comm. Analysis Geometry 10 (2002), 307–339. 5. E. Kuwert and R. Sch¨ atzle, Removability of point singularities of Willmore surfaces, Ann. of Math. 160 (2004), 315–357. 6. J. Langer, A compactness theorem for surfaces with Lp -bounded second fundamental form, Math. Ann. 270 (1985), 223–234. 7. P. Li and S. T. Yau, A new conformal infvariant and its applications to the Willmore conjecture and the first eigenvalue on compact surfaces, Invent. Math. 69 (1982), 269– 291. 8. S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S. Bures-sur-Yvette, 1965. 9. U. F. Mayer and G. Simonett, A numerical scheme for radially symmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow, Interfaces and Free Boundaries 4 (2002), 89–109. 10. L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983), 525–571. 11. G. Simonett, The Willmore flow near spheres, Differential Integral Equations 14 (2001), 1005–1014. 12. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62– 105. 13. T. J. Willmore, Total Curvature in Riemannian Geometry, John Wiley & Sons, New York, 1982. 14. T. J. Willmore, Riemannian Geometry, Oxford Clarendon Press, Oxford, 1996. 15. E. Zeidler, Nonlinear Functional Analysis and Its Applications. I, Springer Verlag, New York, Berlin, Heidelberg, 1990. ´ Paul Verlaine - Metz et CNRS, Laboratoire de Mathe ´matiques Universite ˆ et Applications de Metz, UMR 7122, Bat. A, Ile du Saulcy, 57045 Metz Cedex 1, France E-mail address:
[email protected] ´ 83, Department of Mathematical Analysis, Charles University, Sokolovska 186 75 Praha 8, Czech Republic and Abteilung Angwandte Analysis, Univer¨t Ulm, 89069 Ulm, Germany sita E-mail address:
[email protected] ¨ t Tu ¨bingen, Mathematisches Institut, Auf der Morgenstelle 10, Universita ¨bingen, Germany D-72076 Tu E-mail address:
[email protected]
