Deforming a hypersurface by Gauss curvature and support function

Deforming Convex Hypersurfaces to a Hypersurface with Prescribed Harmonic Mean Curvature

Let F be a smooth convex and positive function defined in A = {x ∈ R : R1 < |x| < R2} satisfying F (x) ≥ nR2 on the sphere |x| = R2 and F (x) ≤ nR1 on the sphere |x| = R1. In this paper, a heat flow method is used to deform convex hypersurfaces in A to a hypersurface whose harmonic mean curvature is the given function F .

Curvature Groups of a Hypersurface

A cochain complex associated with the vector 1-form determined by the first and second fundamental tensors of a hypersurface M in E"+l is introduced. Its cohomology groups HP(M), called curvature groups, are isomorphic with the cohomology groups of M with coefficients in a subsheaf %R of the sheaf S of closed vector fields on M. If M is a minimal variety, the same conclusion is valid with S^ re...

Curvature of a Closed Hypersurface

Deforming the Gauss-Manin Connection

§0. The Gauss-Manin Connection. Let p > 2 be a prime and N ≥ 4. The p-adic modular curve X1(Np) will be denoted X. We let X = W0 ∪W∞ denote the usual decomposition of X as the union of two wide open sets. Hence Wi (i = 0 or ∞) is a wide open neighborhood of the ordinary component Zi containing the cusp i (= 0 or ∞) and the intersection W = W∞ ∩ W0 is the union the supersingular annuli. We have ...

Motion of Hypersurfaces by Gauss Curvature

We consider n-dimensional convex Euclidean hypersurfaces moving with normal velocity proportional to a positive power α of the Gauss curvature. We prove that hypersurfaces contract to points in finite time, and for α ∈ (1/(n + 2], 1/n] we also prove that in the limit the solutions evolve purely by homothetic contraction to the final point. We prove existence and uniqueness of solutions for non-...