Geometry and stability of surfaces with constant anisotropic mean curvature

Koiso, Miyuki; Palmer, Bennett

doi:10.1512/iumj.2005.54.2613

Cited by 66 publications

(84 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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“…In particular, ≡ constant characterizes surfaces which are in equilibrium for the functional Ᏺ subject to a volume constraint; see [Koiso and Palmer 2005]. From our computations, the anisotropic mean curvature is a constant if and only if…”

Section: Generalized Anisotropic Delaunay Surfacesmentioning

confidence: 82%

“…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 which were extensively studied in [Koiso and Palmer 2005]; that derivation was less elementary than the one appearing here.…”

Section: Generalized Anisotropic Delaunay Surfacesmentioning

confidence: 98%

“…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%

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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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“…On the other hand, over the past years, many authors considered geometric problems involving anisotropic mean curvature and higher order mean curvatures (see [4,5,10,13] for instance). The setting is the following.…”

Section: Introductionmentioning

confidence: 99%