Stability of anisotropic capillary surfaces between two parallel planes

Koiso, Miyuki; Palmer, Bennett

doi:10.1007/s00526-005-0336-7

Cited by 27 publications

(43 citation statements)

References 12 publications

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“…We consider the map (2) φ : S n → R n+1 , x → F (x)x + (grad S n F ) x , its image W F = φ(S n ) is a smooth, convex hypersurface in R n+1 called the Wulff shape of F (see [2], [3], [15], [10], [11], [12], [13], [17], [22], [23]). When F ≡ 1, the Wulff shape W F is just S n .…”

Section: Introductionmentioning

confidence: 99%

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

He¹,

Li²,

Ma³

et al. 2009

Indiana Univ. Math. J.

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Abstract. Given a positive function F on S n which satisfies a convexity condition, for 1 ≤ r ≤ n, we define the r-th anisotropic mean curvature function H F r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface without boundary in R n+1 with H F r = constant is the Wulff shape, up to translations and homotheties. In case r = 1, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.

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Section: Introductionmentioning

confidence: 99%

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

He¹,

Li²,

Ma³

et al. 2009

Indiana Univ. Math. J.

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“…whose image W F = φ(S n ) is a smooth, convex hypersurfaces in R n+1 called the Wulff shape of F (for more details concerning the properties of the Wulff shape see, for instance, [9,14,16,17,18,19,27]). We note that, when F ≡ 1 we have that the Wulff shape of F is just the n-dimensional Euclidean sphere S n ⊂ R n+1 .…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Silva¹,

Lima²,

Velásquez³

2018

Publicacions Matemàtiques

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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].

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“…Let us consider the case where (α, β) = (cos τ, sin τ ), that is, W is a smooth, closed convex surface of revolution generated by the curve W in (2-1). The surfaces X (s, τ ) of revolution with constant anisotropic mean curvature were studied in detail in [Koiso and Palmer 2005;2006;2007a;2007b]. We refer to them as anisotropic Delaunay surfaces.…”

Section: Generalized Anisotropic Delaunay Surfacesmentioning

confidence: 99%

“…In view of (2) and (3) above, this should particularly be the case for functionals having a maximum principle, as do the elliptic parametric functionals. In fact, in [Koiso and Palmer 2006;2007a;2007b] the maximum principle has been applied to conclude that solutions of capillary problems for rotationally invariant elliptic functionals with free boundaries on horizontal planes are surfaces of revolution.…”

Section: Introductionmentioning

confidence: 99%

Rolling construction for anisotropic Delaunay surfaces

Koiso¹,

Palmer²

2008

Pacific J. Math.

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Anisotropic Delaunay surfaces are surfaces of revolution that have constant anisotropic mean curvature. We show how the generating curves of such surfaces can be obtained as the trace of a point held in a fixed position relative to a curve that is rolled without slipping along a line. This generalizes the Delaunay's classical construction for surfaces of revolution with constant mean curvature. Our result is given as a corollary of a new geometric description of the rolling curve of a general plane curve. Also, we characterize anisotropic Delaunay curves by using their isothermic self-duality.

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Stability of anisotropic capillary surfaces between two parallel planes

Abstract: We study the stability of capillary surfaces without gravity for anisotropic free surface energies. For a large class of rotationally symmetric energy functionals, it is shown that the only stable equilibria supported on parallel planes are either cylinders or a part of the Wulff shape.

Cited by 27 publications

References 12 publications

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Stability of generalized linear Weingarten hypersurfaces immersed in the Euclidean space

Rolling construction for anisotropic Delaunay surfaces

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