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Geometry of surfaces with constant anisotropic mean curvature (Joint work with Bennett Palmer) M IYUKI KOISO Department of Mathematics, Nara Women’s University
[email protected] Abstract We study the geometry of hypersurfaces which are in equilibrium for a constant coefficient parametric elliptic functional with a volume constraint. The functional serves as a model of an anisotropic surface energy, and each equilibrium hypersurface has a constant anisotropic mean curvature (CAMC hypersurface). We give the first and second variations and a minimizing property of the Gauss map of stable CAMC hypersurfaces. The equilibrium surfaces of revolution (anisotropic Delaunay surfaces) are also discussed. We give a classification of these surfaces and a representation formula for them. Also we characterize them by using their isothermic self-duality. Then, we give a method of constructing CAMC surfaces which are not rotationally symmetric. Moreover, we discuss the uniqueness problem for closed CAMC hypersurfaces. Lastly, we give a new geometric description of the rolling curve of a general plane curve and apply it to the generating curves of anisotropic Delaunay surfaces. This generalizes the classical construction for CMC surfaces due to Delaunay. The results which are discussed here come from recent joint work with Bennett Palmer. Keywords : anisotropic surface energy, anisotropic mean curvature, Delaunay surface, elliptic parametric functional, Wulff shape, rolling curve, roulette
1 Introduction Minimal surfaces and surfaces of constant mean curvature (CMC surfaces) are probably the most fundamental subjects of research in variational problems for surfaces. They have been studied extensively since the latter half of the eighteenth century, and now they have a beautiful theory and are still an active area. They are critical points of the area functional, critical points of the area functional for volumepreserving variations, respectively. Sometimes it is said that they serve as a model of soap films, soap bubbles, respectively, for the free surface energy of liquid is regarded as isotropic and if the liquid is homogeneous, it is proportional to the surface area. However, if the material is a crystalline solid or a liquid crystal, we need to consider an anisotropic surface energy (i.e. one that depends on the direction of the surface). The subject in this article is the geometry of equilibrium surfaces for this type of energy functional. Let F : U ⊂ Sn → R+ be a positive, smooth function. For a smooth, oriented immersed hypersurface (we will simply write hypersurface) X : Σ = Σn → Rn+1 whose Gauss map ν : Σ → Sn is assumed to lie in U, we define the functional Z (1) F (X) := F(ν) dΣ, Σ
where dΣ is the volume form of the induced metric. Such functional is used to model anisotropic surface energies. Applications can be found in many branches of the physical sciences including metallurgy and crystallography ([20], [21]). It is known that, up to translation, there exists a unique absolute minimizer W (V ) of F among all closed ‘hypersurfaces’ in Rn+1 enclosing the same (n + 1)-dimensional volume The author is partially supported by Grant-in-Aid for Scientific Research (C) No. 16540217 of the Japan Society for the Promotion of Science.
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V and it is convex (possibly with singularities) ([19]). Here the term ‘hypersurface’ can be taken in the sense of the boundary of a set of positive Lebesgue measure. Thus W (V ) solves the isoperimetric problem for the functional F . In the special case where F ≡ 1, the absolute minimizer is a round sphere Sn . We will impose a convexity condition on the functional: Denote by DF and D2 F the gradient and Hessian of F on Sn . Then we require that at each point in U the matrix D2 F + F1 is positive definite, where 1 means the identity endomorphism field. The functional appearing in (1) is sometimes called a (constant coefficient) parametric elliptic functional (PEF). It has been extensively studied from the viewpoint of geometric measure theory and convex analysis. However, in differential geometry it had not been studied very much until rather recently. From now on, for simplicity, we will assume that U = Sn holds. Denote by H the mean curvature of X. Then the Euler-Lagrange equation for the functional F for compactly supported volume-preserving variations is divΣ DF − nHF = constant, (2) where DF is considered as a smooth tangent vector field along X by parallel translation in Rn+1 , and A is defined as A := D2 F + F1. In view of (2), the anisotropic mean curvature Λ of X is defined as (cf. [18], [10]) Λ := −divΣ DF + nHF = −traceΣ Adν. If Λ is constant, X is called a hypersurface of constant anisotropic mean curvature (CAMC hypersurface). In the special case where F ≡ 1, Λ = nH holds. Hence, the study of CAMC (hyper)surfaces includes the study of CMC and minimal (hyper)surfaces. The convexity condition implies that the Euler-Lagrange equation “Λ = constant” is absolutely elliptic in the sense of [6]. This implies that a maximum principle analogous to that for CMC surfaces holds. Another interpretation of the convexity condition is the following. Consider the embedding χ : Sn → Rn+1 defined by χ(ν) = DF(ν) + F(ν)ν. Then the convexity condition implies that χ defines a smooth, convex hypersurface in Rn+1 . The hypersurface defined by χ is called the Wulff shape of F. An important result known as Wulff’s theorem, though actually proved by Jean Taylor ([19]), is that the Wulff shape is the absolute minimizer of the functional F among all closed hypersurfaces in Rn+1 enclosing the same (n + 1)-dimensional volume. The anisotropic mean curvature of the Wulff shape is −n with respect to the outward pointing normal, and the unique minimizer W (V ) mentioned above is a homothety of χ(Sn ). In the case where F ≡ 1, the Wulff shape is the sphere with radius 1 and center at the origin. Conversely, for any closed smooth strictly convex hypersurface W , take a point inside of the domain bounded by W as the origin. And set F(ν) := q(ν), where q is the support function of W , that is, q(ν) is the distance between the origin and the tangent hyperplane at the uniquely determined point χ(ν) of W with ν asR the outward pointing unit normal of W at χ(ν). Then, W is the Wulff shape for the functional F [X] = Σ F(ν) dΣ. Since a CAMC hypersurface is a critical point of an anisotropic surface energy, it is natural to ask whether it attains a local minimum of the energy functional or not. A CAMC hypersurface X is said to be stable if the second variation of F is nonnegative for all (n + 1)-dimensional volume-preserving variations of X with compact support. In §2, we give the first and the second variation formulas for the energy functionals, and give some formulas for the stability. Also we give some simple examples of CAMC surfaces. As an application of the second variation, in §3, we give a minimizing property of the Gauss map of stable CAMC hypersurfaces for a generalized Dirichlet integral. Probably the most important examples of constant mean curvature surfaces are the Delaunay surfaces, i.e. the surfaces of revolution. They are not only important examples of CMC surfaces in their own right, but also appear in a fundamental way in the study of more general CMC surfaces. We investigate the analogous class of CAMC surfaces, and refer to a CAMC surface of revolution as an anisotropic Delaunay surface. Such surfaces were studied in detail by the authors in [10] (see also [11] and [12]). The composition of this class of surfaces is strikingly similar to the classical case. When the energy density F is rotationally invariant, they are classified into six classes: plane, Wulff shape (up to translation
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and homothety), right circular cylinders, anisotropic catenoids, anisotropic unduloids and anisotropic nodoids. The anisotropic unduloids are embedded periodic surfaces while the anisotropic nodoids are only immersed and periodic. Anisotropic Delaunay surfaces were used essentially in our study on capillary problems ([11]-[13]). Also, the classification of them is essential for proving our uniqueness result for closed embedded CAMC surfaces which will be given in §7. We will give representation formulas and summarize important results about them in §4. Since any closed convex surface can be considered as a Wulff shape, it is a natural question which rotation surface can be an anisotropic Delaunay surface. Actually, we characterize anisotropic Delaunay surfaces by using their isothermic self-duality (§5). On the other hand, in general, it is difficult to obtain examples of CAMC surfaces except CMC surfaces and surfaces of revolution. We don’t know whether one can find a useful representation formula for general CAMC surfaces or not. However, we obtained a new method of constructing examples of CAMC surfaces whose surface energy is not necessarily rotationally invariant ([14]). It will be useful not only for the study on CAMC surfaces but also for other fields such as crystallography, mathematical biology, and so on. We will show some formulas and some examples about this subject in §6. One of the most fundamental questions of a variational problem is the uniqueness of the solution. In [11], We proved that, for a rotationally symmetric anisotropic surface energy, any embedded closed CAMC surface was (up to translation and homothety) the Wulff shape. In §7, we will give some known results and some conjectures about the uniqueness for closed CAMC surfaces. We should remark that people are interested in this subject also in the study of crystalline curvature flow equations, because closed CAMC surfaces are expected to appear as limits of solutions of these equations. Crystalline curvature flow equations are studied extensively and interdisciplinary, since, for example, it describes a crystal growth phenomena. There is a classical theorem by Delaunay which says that each Delaunay surface except spheres is obtained by rotating a roulette of which the base is a line, the rolling curve is a conic section, and the pole is one of the foci of the rolling curve around the base line. Maybe people are interested in whether this result can be generalized to anisotropic Delaunay surfaces or not. In the last section, we will give a new geometric description of the rolling curve of a general plane curve and apply it to the generating curves of anisotropic Delaunay surfaces. For example, the rolling curve of an anisotropic unduloid or nodoid S is obtained as a type of ‘dual curve’ of a ‘mean curvature profile’ of S. Here, a mean curvature profile of S is a curve whose curvature κ(s) is equal to twice the mean curvature H(s) measured along a meridian of S. As a corollary, we obtain the classical theorem by Delaunay.
2 Variation formulas and the definition of stability For an immersion X : Σ = Σn → Rn+1 , set Z
FΛ0 (X) := F (X) + Λ0V (X) =
Σ
F(ν)dΣ + Λ0V (X),
Λ0 ∈ R,
(3)
where V (X) denotes the algebraic (n + 1)-dimensional volume enclosed by X: V (X) :=
1 n+1
Z Σ
hX, νi dΣ,
where h , i is the canonical inner product in Rn+1 . The condition that X is a critical point of FΛ0 for compactly supported variations is equivalent to the condition that X is a critical point of F for compactly supported volume-preserving variations. However, if we consider the second variation, the situations become different. Let Xε = X + (ξ + ψν)ε + O (ε2 ) be a compactly supported variation of X, where ξ is the tangential component of the variation vector field. Lemma 1 The first variation of F and V are given by Z
Z
∂ε F |ε=0 = −
Σ
Λψ dΣ,
∂εV |ε=0 =
Σ
ψ dΣ,
(4)
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where Λ := −divΣ DF + nHF = −traceΣ Adν is the anisotropic mean curvature of X. Probably we should give some simple examples of CAMC surfaces. Example 1 Using the Legendre transformation (cf. Reilly [18]), one finds that the PEF whose Wulff shape is the ellipsoid x2 y2 z2 + + =1 a2 b2 c2 is given by
Z q
a2 ν21 + b2 ν22 + c2 ν23 dΣ,
F = or in non-parametric form z = z(x, y), Z q
F =c
1 + (azx /c)2 + (bzy /c)2 dxdy.
The transformation x0 = x/a, y0 = y/b, z0 = z/c converts F into the area functional multiplied by abc. It follows that an immersion X = (x, y, z) is a critical point of some FΛ0 if and only if (x0 , y0 , z0 ) is a constant mean curvature immersion. The following results are fundamental for the stability analysis. Proposition 1 ([10]) Assume that the anisotropic mean curvature of X is a constant Λ0 . Let Xε = X + (ξ + ψν)ε + O (ε2 ) be a compactly supported variation of X, where ξ is the tangential component of the variation vector field. Then the first variation of the anisotropic mean curvature Λ is given by ∂ε Λ|ε=0 = L[ψ],
(5)
where L is the self-adjoint operator L[ψ] := div(A∇ψ) + hAdν, dνiψ with A := (D2 F + F1)|ν . The second variation of FΛ0 is given by Z
∂2εε FΛ0 |ε=0 = −
Σ
ψL[ψ]dΣ.
(6)
If Xε satisfies a further assumption that it is volume-preserving, then Z
∂2εε FΛ0 |ε=0 = ∂2εε F |ε=0 = −
Σ
ψL[ψ]dΣ.
(7)
Definition 1 Assume that X : Σ → Rn+1 is an immersion with Zconstant anisotropic mean curvature Λ. A relatively compact subdomain Ω ⊂ Σ is said to be stable if − ψ with compact support in Ω satisfying
Σ
ψL[ψ]dΣ ≥ 0 holds for all C∞ function
Z
Λ
Ω
ψdΣ = 0.
Z
A domain Ω is said to be strongly stable if − support in Ω.
Σ
ψL[ψ]dΣ ≥ 0 holds for all C∞ function ψ with compact
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Note that Ω is strongly stable if and only if the first eigenvalue λ1 (L, Ω) of the eigenvalue problem L[ψ] = −λψ,
ψ ∈ Cc∞ (Ω)
is nonnegative. Since a one parameter family of translations Xε = X + εC, where C is a constant vector, leaves the anisotropic mean curvature invariant, from (5), the normal component of the variation vector field νC := hν,Ci solves L[νC ] = 0. Therefore, L[ν j ] = 0,
j = 1, · · · , n + 1,
(8)
here we write ν = (ν1 , ν2 , · · · , νn+1 ) : Σn → Sn ⊂ Rn+1 .
3 Minimizing property of the Gauss map Let (Σn , ∂Σ) be a smooth manifold with smooth boundary. Let X : Σ = Σn → Rn+1 be an immersion with constant anisotropic mean curvature, and let ν be its Gauss map. Then, ν is a critical point of the functional Z E[ f ] :=
Σ
hA∇ f , ∇ f i dΣ
in the space { f : Σ → Sn ; f |∂Σ = ν|∂Σ }. In fact, we can prove the following result. Theorem 1 ([10]) Let (Σn , ∂Σ) be a smooth manifold with smooth boundary and let X : Σ → Rn+1 be an immersion with constant anisotropic mean curvature. If X is strongly stable, then the Gauss map ν has the following minimizing property: for every C∞ map f : Σ → Sn with f ≡ ν on ∂Σ, there holds E[ν] ≤ E[ f ].
(9)
If X is stable, then (9) holds for all C∞ maps f : Σ → Sn such that f ≡ ν on ∂Σ and Z Σ
f − ν dΣ = 0,
i.e. f and ν have the same center of mass. We make note of the following simple corollary. Corollary 1 ([10]) Let X : Σ → Rn+1 be an immersed compact CMC (possibly minimal) hypersurface with boundary with λ1 (L, Σ) ≥ 0. Then the Gauss map ν is the absolute minimizer of the Dirichlet energy among all sufficiently smooth maps into Sn with the same boundary values as the map ν. Here we give the idea of the proof of Theorem 1. Set λ1 := λ1 (L, Σ). Represent ν, f as ν = (ν1 , · · · , νn+1 ), f = ( f1 , · · · , fn+1 ) : Σ → Sn ⊂ Rn+1 . Then, Z
Z
λ1
Σ
(ν j − f j )2 dΣ ≤
Σ
hA∇(ν j − f j ), ∇(ν j − f j )i − hA∇ν, ∇νi(ν j − f j )2 dΣ
holds. Expand this out and compute, using that the mappings are into a sphere and using the selfadjointness of A and (8).
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4 Anisotropic Delaunay surfaces In this section, we restrict ourselves to the two-dimensional case. Denote by µ1 , µ2 the principal curvatures of the Wulff shape W ⊂ R3 with respect to the inward pointing normal. Let X : Σ = Σ2 → R3 be an immersion with constant anisotropic mean curvature Λ, and let ν : Σ → S2 its Gauss map. For p ∈ Σ, ¯ there exists a uniquely determined point G(p) in W such that ν(p) coincides with the outward pointing ¯ unit normal to W at G(p). We call this map G¯ : Σ → W the anisotropic Gauss map of X. Let C be a curve in X which passes a point p. Assume that the direction of C at p coincides with the ¯ principal direction of W which corresponds to µ j at the point G(p). Denote by h j j the normal curvature of C at p. Then, it is shown that Λ = h11 /µ1 + h22 /µ2 (10) holds. Now assume that F is rotationally invariant, say F = F(ν3 ). Then the Wulff shape W is also rotationally invariant with respect to the third axis, and the principal directions of W are given by e1 := (0, 0, 1) − ν3 ν,
e2 := ν × e1 .
The principal curvature µ j corresponding to these directions are given by µ ¶ 1 − ν23 1 0 1 1 1 1 + , = (1 − ν23 )F 00 + =− = F − ν3 F 0 . µ1 µ2 ν3 µ2 µ2 µ2
(11)
Note that µ1 is the principal curvature of W in the direction of its generating curve, and µ2 is that of W in the direction of rotation. µ1 and µ2 are positive functions on S2 which depend only on ν3 . Let
χ(σ, θ) = (u(σ)eiθ , v(σ))
be a parametrization of the Wulff shape W , where (u(σ), v(σ)) is the arc length parametrization of the generating curve. We have identified R3 with C × R in the formula above. We may extend (u(σ), v(σ)) so that it is defined for all real number σ. In this case, (u(σ), v(σ)) represents the section of W by (x1 , x3 )-plane. Consider an anisotropic Delaunay surface Σ parameterized by X(s, θ) = (x(s)eiθ , z(s)) , where (x(s), z(s)) is the arc length parameterization of the generating curve, and x(s) ≥ 0 holds for all s. The Gauss map of the surface X is given by ν = (z0 (s)eiθ , −x0 (s)). We choose the orientation of the generating curve so that ν points “outward” from the surface. At ¯ the outward pointing unit the corresponding points of X and W through the anisotropic Gauss map G, normals must agree and we have x0 = uσ , z0 = vσ . (12) Inserting (11) to (10), and integrating the equation, we obtain that the profile curve (x, z) satisfies the equation 2µ2 −1 xz0 + Λx2 = c , (13) where c is a real constant called the flux parameter. Since W is a surface of revolution, we have µ2 = µ2 (ν3 ) = µ2 (−uσ ) = µ2 (−x0 ) by (12). Computing the principal curvature −µ2 = −vσ /u, (13) can be expressed as 2ux + Λx2 = c . (14) The orientation of an anisotropic Delaunay surface may be chosen so that Λ ≤ 0 holds and then the anisotropic Delaunay surfaces fall into six cases as follows: • (I-1) Λ = 0 and c = 0: horizontal plane.
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• (I-2) Λ = 0 and c 6= 0: anisotropic catenoid. • (II-1) Λ < 0 and c = 0: Wulff shape (up to vertical translation and homothety). • (II-2) Λ < 0 and c = ((µ2 |ν3 =0 )2 |Λ|)−1 : cylinder of radius (µ2 |ν3 =0 |Λ|)−1 . • (II-3) Λ < 0 and ((µ2 |ν3 =0 )2 |Λ|)−1 > c > 0: anisotropic unduloid. • (II-4) Λ < 0 and c < 0: anisotropic nodoid. The generating curve of a surface in the cases (I-2), (II-3) and (II-4) will be called an anisotropic catenary, an anisotropic undulary and an anisotropic nodary, respectively. Any surface in each case above is complete, and it has similar properties to the corresponding CMC surface in the sense of the following Lemma. Theorem 2 ([10], [11], [12]) (i) The generating curve C : (x(s), z(s)) of an anisotropic catenoid is a graph over the whole z-axis, and z0 (s) 6= 0 for all s. C is perpendicular to the horizontal line at a unique point. (ii) Let (x(s), z(s)), (x ≥ 0), be the generating curve of an anisotropic unduloid or an anisotropic nodoid. Then, there is a unique local maximum B and a unique local minimum N > 0 of x, which we will call a bulge and a neck respectively. (iii) The generating curve C : (x(s), z(s)) of an anisotropic unduloid is a graph over the z-axis, and z0 (s) > 0 for all s. C is a periodic curve with respect to the vertical translation, and the region from a neck to the next neck (and/or a bulge to the next bulge) gives one period. Therefore, C has a unique p inflection point (x, z) between each neck and the next bulge, which satisfies x = c/(−Λ). (iv) The curvature of the generating curve C of an anisotropic nodoid has a definite sign. C is a nonembedding periodic curve with respect to the vertical translation. The region from a neck to the next neck (and/or a bulge to the next bulge) gives one period. At the end of this article, there are some pictures of anisotropic Delaunay surfaces (Figures 2-4) whose Wulff shape is determined by (x2 + y2 )2 + z4 = 1 (Figure 1). See also Figures 3-6 in another article [16] in the same conference proceedings. The following representation formula for the profile curves was used essentially in our study on CAMC surfaces in [11]-[14]. Theorem 3 ([10]) Let W be the Wulff shape of a rotationally symmetric anisotropic surface energy F . Let σ 7→ (u(σ), v(σ)), σ ∈ (−∞, ∞), be the profile curve of W , where σ is the arc length. Then µ−1 2 vσ − u = 0 holds. Let X(s, θ) = (x(s)eiθ , z(s)) be a surface with constant anisotropic mean curvature Λ ≤ 0, and let the Gauss map of X coincide with that of W at s = s(σ). Then X is given as follows. (i) When X is an anisotropic catenoid, x = c/(2u)
(15)
√ u ± u2 + Λc x= −Λ
(16)
for some nonzero constant c. (ii) When X is an anisotropic unduloid,
for some constants c > 0 and Λ < 0, where x = x(u(σ)) is defined in {σ|u ≥
√
−Λc}.
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(iii) When X is an anisotropic nodoid, √ u + u2 + Λc x= −Λ
(17)
for some constants c < 0 and Λ < 0, where x = x(u(σ)) is defined in {−∞ < σ < ∞}. In all cases above, z is given by z=
Z u
vu xu du.
(18)
Conversely, for a Wulff shape W defined as above, define x and z as in (i) – (iii) and (18). Then X(s, θ) = (x(s)eiθ , z(s)) is an anisotropic Delaunay surface which satisfies 2 2µ−1 2 zs x + Λx = c,
where s is the arc length of (x, z), and Λ is supposed to be zero for Case (i). Moreover, X has the same regularity as that of W . √ Remark 1 In (ii) in Theorem 3, x = (−Λ)−1 (u + u2 + Λc) gives the part of the anisotropic unduloid √ whose Gaussian curvature is positive (i.e. the convex part), while x = (−Λ)−1 (u − u2 + Λc) gives the part of the anisotropic unduloid whose Gaussian curvature is negative. Remark 2 In (iii) in Theorem 3, u > 0 corresponds to the part of the anisotropic nodoid whose Gaussian curvature is positive (i.e. the convex part), while u < 0 gives the part of the anisotropic nodoid whose Gaussian curvature is negative. Proposition 2 ([14]) For any anisotropic Delaunay surface X : Σ → R3 , the anisotropic Gauss map G¯ : Σ → W is harmonic. In particular, when the surface is an anisotropic catenoid or the Wulff shape, G¯ is ± holomorphic.
5 Characterization of anisotropic Delaunay curves What surfaces of revolution arise as anisotropic Delaunay surfaces? Here, using isothermic duality, we characterize the surfaces which can arise as anisotropic unduloids and nodoids without making explicit reference to the anisotropic energy functional. Let Ω(s) = (x(s), z(s)),
x(s) > 0
be a smooth curve with arc length s. We consider the surface (x(s)eiθ , z(s)) of revolution generated by Ω. First recall that a surface S is called isothermic if, away from umbilic points, its lines of curvature are given as the level sets of a pair of locally defined conjugate harmonic functions, and it is clear that a surface of revolution is isothermic. According to a classical theorem of Bour and Christofell ([2]), ˜ defined up to homothety and translation, to each isothermic surface corresponds an isothermic dual S, with the property that S and S˜ are anticonformal and they share the same Gauss map. The construction of the isothermic dual involves integration and is therefore only well defined, in general, over simply connected domains. In the case of a surface of revolution, it is shown that the duality is global in the following way. ˜ Define a curve Ω(s) = (x(s), ˜ z˜(s)) as x(s) ˜ :=
a , x(s)
z˜(s) := −a
Z s 0 z 0
x2
ds,
(19)
iθ , z˜(s)) of revolution generated by Ω ˜ gives the where a is a nonzero constant. Then the surface (x(s)e ˜ iθ isothermic dual of the surface (x(s)e , z(s)).
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Now let us assume that the curve Ω is periodic with period L in the following sense: x and z satisfy x(s + L) = x(s),
∀s ∈ R,
z(s + L) − z(s) = z(L) − z(0),
∀s ∈ R.
We assume that one period of Ω contains a unique local maximum and a unique local minimum of x, which will be called a bulge and a neck of Ω, respectively. We also assume that Ω satisfies either the following condition (I) or (II): (I) There is only one inflection point between a bulge and the next neck, and there is no zero of z0 . (II) There is no inflection point, and there is only one zero of z0 between a bulge and the next neck. We may assume that s = 0 corresponds to a bulge, s = −l1 , l2 (l1 , l2 > 0) correspond to the next necks, and z(0) = 0. We denote by −sI , sJ (−l1 < −sI < 0 < sJ < l2 ) the unique point in (I) or (II) above. ˜ defined by (19). We choose a positive constant as a if Ω satisfies (I), and a We consider the curve Ω negative constant as a if Ω satisfies (II). Theorem 4 ([14]) If
˜ Ω(R) = Ω(R)
(20)
holds up to vertical translation and reflection with respect to the vertical axis (the third axis), then Ω is R an anisotropic undulary or nodary for some anisotropic surface energy F = F(ν3 )dΣ with rotationally symmetric energy integrand. R
Conversely, if Ω is an anisotropic undulary or nodary for an anisotropic surface energy F = F(ν3 ) dΣ with rotationally symmetric energy integrand, then Ω is periodic, either (I) or (II) holds, and Ω(R) = ˜ Ω(R) holds for some constant a 6= 0 up to vertical translation.
6 Generalized anisotropic Delaunay surfaces In this section we give a description of some equilibrium surfaces for functionals whose Wulff shape may not be rotationally symmetric. The Wulff shape is assumed to have the property that all of its intersections with horizontal planes are mutually homothetic. In the case when the Wulff shape is a surface of revolution, the construction reduces to that of the anisotropic Delaunay surfaces. Let ΩW : (u(σ), v(σ)),
−2L1 ≤ σ ≤ 2L2
(21)
be a closed convex curve parametrized by arc length σ which is symmetric with respect to the v-axis. We may assume that u ≥ 0,
vσ ≥ 0,
−L1 ≤ σ ≤ L2 ,
(22)
u < 0,
vσ < 0,
σ ∈ [−2L1 , −L1 ) ∪ (L2 , 2L2 ]
(23)
hold. Also we may assume that u and v0 have zeros only at σ = −L1 , L2 and u0 and v have zeros only at σ = 0, −2L1 , 2L2 . Let C : τ → (α(τ), β(τ)) be a closed convex curve parameterized by arc length in the plane. We assume that the origin is inside the domain bounded by C. Consider the surface W given by χ(σ, τ) = (u(σ)α(τ), u(σ)β(τ), v(σ)) . When (α, β) = (cos τ, sin τ), this gives a surface of revolution. In general, W is a convex surface such that all the curves obtained by intersecting W with planes x3 = constant are homothetic to each other. For a suitable plane curve (x(s), z(s)) parameterized by arc length s with x > 0, we have the surface Σ defined by X(s, τ) = (x(s)α(τ), x(s)β(τ), z(s)) . (24) The normals to the two surfaces agree exactly when xs = uσ ,
zs = vσ
(25)
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hold if z0 ≥ 0 holds, and when xs = −uσ ,
zs = −vσ
(26)
z0
hold if < 0 holds. Recall that the Gauss map of a convex surface is a diffeomorphism onto the 2sphere. The assignment to a point X(s, τ) on Σ the point χ(σ, τ) on W where (25) or (26) holds defines a map G¯ : Σ → W which we call the anisotropic Gauss map of X (cf. §4). It is shown that
Λ = −trace d G¯ = −(du/dx + u/x)
(27)
holds. The equation (27) is the same as (xu)x = −Λx. Integrating this equation gives
ux = −Λ(x2 /2) + c/2 ,
(28)
where c is a constant. The equation (28) is the same as (14). Therefore, if we define x and z as in (15) (17) and (18) in Theorem 3, then X(s, τ) = (x(s)α(τ), x(s)β(τ), z(s)) defines a surface with constant anisotropic mean curvature Λ for the Wulff shape given by χ(σ, τ) = (u(σ)α(τ), u(σ)β(τ), v(σ)). We call these surfaces a generalized anisotropic catenoid, a generalized anisotropic unduloid, and a generalized anisotropic nodoid, respectively. At the end of this article, there are some pictures of examples of these surfaces (Figures 6-8) whose Wulff shape is determined by x4 + y4 + z4 = 1 (Figure 5). The Gielis formula ([3]) r = {|(1/a) cos(
mθ n2 mθ )| + |(1/b) sin( )|n3 }−n1 4 4
(29)
can be used to generate a large number of interesting closed planer curves with symmetries. Here let us consider only the cases where a = 1 = b holds, so we will omit these parameters. When such a curve is represented parametrically, we will denote it by G(θ, m, n1 , n2 , n3 ). Two such curves G(σ, m, n1 , n2 , n3 ) = (u(σ), v(σ)), G(τ, M, N1 , N2 , N3 ) = (α(τ), β(τ)) can be used in the formula χ(σ, τ) = (u(σ)α(τ), u(σ)β(τ), v(σ)) to produce a closed surface. When each of the curves is convex and the curve (u(σ), v(σ)) is symmetric with respect to the v-axis, χ defines an embedded closed convex surface. We can regard this closed surface as the Wulff shape W and construct generalized anisotropic Delaunay surfaces ([15]).
7 Uniqueness problem for closed CAMC surfaces The following uniqueness result is known. Theorem 5 (B. Palmer [17]) Any closed stable hypersurface with nonzero constant anisotropic mean curvature is, up to translation and homothety, the Wulff shape. Let us consider the two-dimensional case. If the anisotropic surface energy F is rotationally invariant, then, by using the maximum principle and the Alexandrov reflection method, we can show that any embedded closed CAMC surface for F has the same rotational symmetry as F. Therefore, by virtue of the classification of anisotropic Delaunay surfaces (§4), we see that this surface is, up to translation and homothety, the Wulff shape. Hence we obtain: Theorem 6 ([11]) Assume that F is (therefore, W is) rotationally symmetric. If X is a closed embedded surface of constant anisotropic mean curvature, then X(Σ) is, up to translation and homothety, the Wulff shape.
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Recently, Y. He and H. Li [4] proved the following result: If X : Σn → Rn+1 is a closed hypersurface with constant anisotropic mean curvature for which hX, νi has a fixed sign, then X(Σ) is, up to translation and homothety, the Wulff shape. We conjecture that if either X : Σ = Σn → Rn+1 is a closed embedded CAMC hypersurface or X : Σ → R3 is a closed CAMC surface with genus zero, then X(Σ) is, up to translation and homothety, the Wulff shape.
8 Rolling construction for general curves and its application to anisotropic Delaunay surfaces We consider a smooth curve Ω(s) = (x(s), z(s)),
x > 0,
l1 ≤ s ≤ l2 ,
(l1 ≤ 0 < l2 ), represented by the arc length s. The curve Ω will always be regarded as the generating curve of a surface of revolution S : (x(s)eiθ , z(s)). According to [7], there exists a curve Γ : r = r(θ) such that Ω(s) is obtained as the trace of a particular point, called the pole, attached to Γ (it is not necessary that the pole is a point of Γ) when Γ rolls without slipping along the z-axis. Γ is called the rolling curve, and Ω is called the roulette. In this section, we will give a new geometric description of the rolling curve of a general plane curve. Then, we will apply it to anisotropic Delaunay surfaces. First we define the mean curvature profile of a surface of revolution. Let S be a surface of revolution generated by a smooth curve Ω(s) = (x(s), z(s)) parameterized by arc length. We assume that either x(s) > 0 (∀s) or x(s) < 0 (∀s) holds. Denote by HS (s) the mean curvature of S along the meridian through Ω(s) with respect to the inward (resp. outward) pointing normal, if xz0 > 0 (resp. xz0 < 0). By the mean curvature profile of S, we mean the plane curve CS (s) = ( f (s), g(s)) defined by the properties: (A) s is also the arc length parameter of CS . (B) The curvature κCS (s) of CS is given by κCS (s) = 2HS (s). Note that the mean curvature profile is only determined up to rigid motion. The mean curvature profile was used extensively by Kenmotsu to study surfaces of revolution with prescribed mean curvature ([8], [9]). It is elementary that, up to rotation, the mean curvature profile ( f , g) is given by the following formulas: f (s) :=
Z s 0
cos θ(s1 ) ds1 − c1 ,
where θ(s) := 2
g(s) :=
Z s 0
Z s 0
sin θ(s1 ) ds1 − c2 ,
HS (s) ds,
(30)
(31)
and c1 and c2 are constants. In [8], Kenmotsu showed that one can determine constants c1 , c2 so that the following equalities hold: p x = (sgn x) f 2 + g2 ,
f g0 − f 0 g , z0 = (sgn x) p f 2 + g2
(32)
where sgnx is the sign of x. The first equality in (32) can be expressed as follows: (C) The distance between a point of Ω and the z axis is equal to the distance between the corresponding point of CS and the origin. (A), (B), (C) determine the curve CS (s) = ( f (s), g(s)) up to rotation. We can characterize the mean curvature profile in another way as follows:
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Proposition 3 ([14]) Let CS (s) = ( f (s), g(s)) be a plane curve defined by the above properties (A) and (C). Then, (B) is automatically satisfied (up to sign). The curve CS (s) = ( f (s), g(s)) is determined up to rotation around the origin. When ( f , g) satisfies (A), (B), and (C), we call the curve ( f , g) the mean curvature profile associated with S. Next we define the dual curve of a plane curve. Let γ be a smooth arc in the plane with unit normal N. Let p := hγ, Ni be the support function which we can consider to also be (locally) a function on a subset of the unit circle S1 . We assume that p has at most isolated zeros. We define the dual curve of γ with respect to the origin by γ∗ := p−1 N. Theorem 7 ([14]) Let S be a smooth surface of revolution generated by the curve Ω(s) = (x(s), z(s)),
x > 0,
l1 ≤ s ≤ l2 ,
(l1 ≤ 0 < l2 ), represented by the arc length s. Assume that the surface S : (x(s)eiθ , z(s)) restricted to Ω has at most isolated umbilics. Assume also that z0 has at most isolated zeros. And regard Γ as a curve in the projective plane if z0 has isolated zeros. Then the rolling curve Γ of Ω is a piecewise C1 ˜ = (x, curve which is smooth away from the umbilics of S. Define Ω ˜ z˜) by (19). And let CS˜ be the mean ˜ Then, the rolling curve Γ ˜ curvature profile associated with the surface S of revolution generated by Ω. of Ω satisfies, up to homothety, Γ = CS∗˜ , i.e. Γ is (up to homothety) dual curve of CS˜ = ( f , g) with respect to the origin of the coordinate plane. Moreover, the pole of the rolling construction is the origin. Furthermore, Γ is a closed curve if x(l1 ) x(l2 ) = , z0 (l1 ) z0 (l2 ) and θ(l2 ) − θ(l1 ) = 2nπ,
n∈Z
hold. Remark 3 In [9] Kenmotsu studied periodic surfaces of revolution with a prescribed mean curvature function. He classified the periodic surfaces of revolution into two classes. One class, which appears in Theorem 2 of [9], has a mean curvature function for a one parameter family of periodic surfaces of revolution. In the class in Theorem 3 of the same paper, each mean curvature function has only an isolated periodic surface of revolution. It can be shown that the surfaces appearing in Theorem 2 are exactly those periodic surfaces whose rolling curves is closed while those appearing in Theorem 3 are exactly those periodic surfaces whose rolling curve is not closed. We will now briefly discuss how this theorem apply to the anisotropic Delaunay surfaces. If S is the Wulff shape, then S˜ is an anisotropic catenoid, while if S is an anisotropic catenoid, then S˜ is, up to reflection with respect to the vertical axis, a homothety of the Wulff shape. If S is an anisotropic unduloid (resp. nodoid), by virtue of Theorem 4, S˜ is, up to reflection with respect to the vertical axis, a homothety of S. Therefore, the construction of the rolling curves is rather simple for anisotropic Delaunay surfaces. In the special case where S is a Delaunay surface except spheres, the mean curvature profile CS˜ associated with S˜ is a circle, and hence the dual curve CS∗˜ is a conic section, that is, we obtain the classical theorem of Delaunay.
References [1] C. Delaunay. Sur la surface de revolution dont la courbure moyenne est constante. J. Math. Pures et Appl. (1) 6 (1841), 309–320. [2] L. P. Eisenhart. A Treatise on the Differential Geometry of Curves and Surfaces. Dover Publications, Inc., New York, 1960.
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[3] J. Gielis. A generic geometric transformation that unifies a large range of natural and abstract shapes. American Journal of Botany 90 (3) (2003), Invited Special Paper, 333–338. [4] Y. He and H. Li. Integral formula of Minkowski type and new characterization of the Wulff shape. arXiv:math.DG/0703187v1, 7 Mar 2007. [5] H. Hopf. Differential Geometry in the Large, Second edition. Lecture Notes in Mathematics 1000, Springer-Verlag, Berlin, 1989. [6] H. B. Jenkins. On two-dimensional variational problems in parametric form. Arch. Rational. Mech. Anal. 8 (1961), 181–206. [7] W.-Y. Hsiang and W. C. Yu. A generalization of a theorem of Delaunay. J. Differential Geom. 16 (1981), 161–177. [8] K. Kenmotsu. Surfaces of revolution with prescribed mean curvature. Tohoku Math. J. 32 (1980), 147–153. [9] K. Kenmotsu. Surfaces of revolution with periodic mean curvature. Osaka J. Math. 40 (2003), 687–696. [10] M. Koiso and B. Palmer. Geometry and stability of surfaces with constant anisotropic mean curvature. Indiana University Mathematics Journal 54 (2005), 1817–1852. [11] M. Koiso and B. Palmer. Stability of anisotropic capillary surfaces between two parallel planes. Calculus of Variations and Partial Differential Equations 25 (2006), 275–298. [12] M. Koiso and B. Palmer. Anisotropic capillary surfaces with wetting energy. Calculus of Variations and Partial Differential Equations 29 (2007), 295–345. [13] M. Koiso and B. Palmer. Uniqueness theorems for stable anisotropic capillary surfaces. To appear in SIAM Journal on Mathematical Analysis. [14] M. Koiso and B. Palmer. Rolling construction for anisotropic Delaunay surfaces. Preprint, 2007. [15] M. Koiso and B. Palmer. Equilibria for anisotropic surface energies and the Gielis formula. Preprint, 2007. [16] M. Koiso and B. Palmer. Anisotropic capillary surfaces. Proceedings of “Symposium on the Differential Geometry of Submanifolds — July 2007, Valenciennes, France —”, 2007. [17] B. Palmer. Stability of the Wulff shape. Proc. Amer. Math. Soc. 126 (1998), 3661–3667. [18] R. C. Reilly. The relative differential geometry of nonparametric hypersurfaces. Duke Math. J. 43 (1976), 705–721. [19] J. E. Taylor. Crystalline variational problems. Bull. Amer. Math. Soc. 84 (1978), 568–588. [20] W. L. Winterbottom. Equilibrium shape of a small particle in contact with a foreign substrate. Acta Metallurgica 15 (1967), 303–310. [21] G. Wulff. Zur Frage der Geschwindigkeit des Wachsthums und der Aufl¨osung der Krystallfl¨achen. Zeitschrift f¨ur Krystallographie und Mineralogie 34 (1901), 449–530.
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Figure 1: Wulff shape determined by (x2 + y2 )2 + z4 = 1
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Figure 2: An end of an anisotropic catenoid for the Wulff shape in Figure 1
Figure 4: A half period of an anisotropic nodoid for the Wulff shape in Figure 1. Λ = −1, c = −20 Figure 3: A half period of an anisotropic unduloid for the Wulff shape in Figure 1. Λ = −1/2, c = 1
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Figure 5: Wulff shape determined by x4 + y4 + z4 = 1
Figure 7: A part of a generalized anisotropic unduloid for the Wulff shape in Figure 5
Figure 6: A part of a generalized anisotropic catenoid for the Wulff shape in Figure 5
Figure 8: A part of a generalized anisotropic nodoid for the Wulff shape in Figure 5
