Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Abstract

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.

An overdetermined problem for the anisotropic capacity

“…Many other authors developed the related theory in several settings, considering both analytic aspects (see [4,7,8,11,[13][14][15][16]27,28]) and geometric points of view (see [9,12,19]). In this paper we will study the anisotropic capacity problem (6) and the associated overdetermined problem (6)- (7).…”

“…Following [2, formulae (3.3), (3.9)] (see also [1,19,23,28]), the anisotropic mean curvature of ∂ , which we shall denote by M H , is defined by…”

The Matzoh Ball Soup Problem: A complete characterization

“…The proof of the Theorem 3.1 relies on the results obtained in [3,4], which generalize the classical ones of Levi-Civita and Segre [7,18]. In this new setting, the metric defined on the submanifolds (the level sets of the function ϕ) is given by an anisotropic (non constant) operator.…”

Upper Bounds for the First Eigenvalue of the Laplacian of Hypersurfaces in terms of Anisotropic Mean Curvatures

“…For instance, in [6], He, Li, Ma and Ge proved an anisotropic version of the well-known Alexandrov theorem [1,12]. Namely, they proved that a compact manifold without boundary M n embedded into R n+1 with constant anisotropic r-th mean curvature is the Wulff shape.…”