Conformal fields and the stability of leaves with constant higher order mean curvature

Conformal Structures and Necksizes of Embedded Constant Mean Curvature Surfaces

Let M = Mg,k denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [15]. Let P = Pg,k = Rg,k × R+ be the space of parabolic structures over Riemann surfaces of genus g with k (marked) punctures, the real analytic structure coming from the 3g− 3+ k loc...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Stability and Bifurcation for Surfaces with Constant Mean Curvature

We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in R, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids inR.

Stability of Hypersurfaces with Constant R-th Anisotropic Mean Curvature

Given a positive function F on S which satisfies a convexity condition, we define the r-th anisotropic mean curvature function H r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. Let X : M → R be an n-dimensional closed hypersurface with H r+1 =constant, for some r with 0 ≤ r ≤ n− 1, which is a critical point for a variational problem. We show tha...

Spacelike hypersurfaces in de Sitter space with constant higher-order mean curvature

ing from (2.6), we obtain that ∫ M ( H1Hr −Hr+1 〈N ,a〉dV = 0. (3.1) We know from Newton inequality [2] that Hr−1Hr+1 ≤ H2 r , where the equality implies that k1 = ··· = kn. Hence Hr−1 ( H1Hr −Hr+1 ≥Hr ( H1Hr−1−Hr ) . (3.2) It derives from Lemma 2.1 that 0≤H1/r r ≤H1/r−1 r−1 ≤ ··· ≤H1/2 2 ≤H1. (3.3) Thus we conclude that Hr−1 ( H1Hr −Hr+1 ≥Hr ( H1Hr1 −Hr ≥ 0, (3.4) and if r ≥ 2, the equalities h...