Stability of the Wulff shape

Palmer, Bennett

doi:10.1090/s0002-9939-98-04641-3

Cited by 60 publications

(41 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…), any closed CAMC hypersurface X is also smooth and the above expectation was already proved. In fact, if X satisfies one of (I)-(III), it is a homothety of the Wulff shape (which was proved by the following papers: For (I), Reference [14] for γ ≡ 1 and Reference [5] for general γ; for (II), Reference [15] for γ ≡ 1 and Reference [8] for general γ; and for (III), Reference [16] for γ ≡ 1 and Reference [7] for general γ). However, the situation is not the same for more general γ and/or W γ .…”

Section: (I)mentioning

confidence: 89%

See 1 more Smart Citation

Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

Koiso

2019

MCA

View full text Add to dashboard Cite

We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered hypersurface, which was introduced to model the surface tension of a small crystal. The purpose of this paper is two-fold. First, we give uniqueness and nonuniqueness results for closed equilibria under weaker assumptions on the regularity of both considered hypersurfaces and the anisotropic surface energy density than previous works and apply the results to the anisotropic mean curvature flow. This part is an announcement of two forthcoming papers by the author. Second, we give a new uniqueness result for stable anisotropic capillary surfaces in a wedge in the three-dimensional Euclidean space.

show abstract

Section: (I)mentioning

confidence: 89%

“…These assumptions on the regularity are weaker than any previous works that studied variational problems of anisotropic surface energies in differential geometry (cf. References [4][5][6][7][8][9]). Under such a weak regularity assumption, we concentrate upon the problem of uniqueness for closed equilibrium hypersurfaces.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

Koiso

2019

MCA

View full text Add to dashboard Cite

show abstract

“…Now, we will quote well known result which will be used later (for a proof, see He and Li [13,15] or Palmer [23]). …”

Section: Key Lemmasmentioning

confidence: 99%

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Silva¹,

Lima²,

Velásquez³

2018

Publicacions Matemàtiques

View full text Add to dashboard Cite

Given a positive function F defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces M n immersed in the Euclidean space R n+1 , the so-called k-th anisotropic mean curvatures H F k , 0 ≤ k ≤ n. For fixed 0 ≤ r ≤ s ≤ n, a hypersurface M n of R n+1 is said to be (r, s, F )-linear Weingarten when its k-th anisotropic mean curvatures H F k , r ≤ k ≤ s, are linearly related. In this setting, we establish the concept of stability concerning closed (r, s, F )-linear Weingarten hypersurfaces immersed in R n+1 and, afterwards, we prove that such a hypersurface is stable if, and only if, up to translations and homotheties, it is the Wulff shape of F . For r = s and F ≡ 1, our results amount to the standard stability studied, for instance, by Alencar-do Carmo-Rosenberg [1].

show abstract

“…its image W F = φ(S n ) is a smooth, convex hypersurface in R n+1 called the Wulff shape of F (see Refs. 8,9,12,22). Now let X : M → R n+1 be a smooth immersion of a compact, orientable hypersurface without boundary.…”

Section: Variations Of Some Parametric Elliptic Functionalmentioning

confidence: 99%

“…Let X : M → R n+1be a smooth immersion of an oriented closed, stable critical point of A r for all variations of X preserving A r−1 . Then, up to translations and homotheties, X(M ) is the Wulff shape.In Ref 22,. B. Palmer considered a variational problem of the functional A 0 restricted to those hypersurfaces preserving the enclosed volume.…”

mentioning

confidence: 99%

Some variational problems in geometry of submanifolds

2008

Differential Geometry and Its Applications

View full text Add to dashboard Cite

show abstract

Stability of the Wulff shape

Abstract: Abstract. We consider the functional of a hypersurface, given by a convex elliptic integrand with a volume constraint. We show that, up to homothety and translation, the only closed, oriented, stable critical point is the Wulff shape.

Cited by 60 publications

References 5 publications

Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Some variational problems in geometry of submanifolds

Contact Info

Product

Resources

About