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THE CONVEX HULL PROPERTY OF NONCOMPACT HYPERSURFACES WITH POSITIVE CURVATURE
 

Summary: THE CONVEX HULL PROPERTY OF NONCOMPACT
HYPERSURFACES WITH POSITIVE CURVATURE
STEPHANIE ALEXANDER AND MOHAMMAD GHOMI
Abstract. We prove that in Euclidean space Rn+1
, every metrically complete,
positively curved immersed hypersurface M, with compact boundary M, lies
outside the convex hull of M provided that M is embedded on the boundary
of a convex body and n > 2. For n = 2, on the other hand, we construct examples
which contradict this property.
1. Introduction
As we showed recently [2], a compact, immersed positively curved hypersurface
of Euclidean space satisfies a Convex Hull Property (CHP), dual to the classical one
for nonpositively curved hypersurfaces [13]. Our CHP states that the hypersurface
lies outside the convex hull of its boundary, provided the boundary satisfies certain
required conditions, e.g., it is embedded on the boundary of a convex body. A proof
of a version of this fact has also been obtained by Guan and Spruck [9] via their work
on Monge-Amp´ere equations. In this note we study generalization of CHP to non-
compact, (metrically) complete hypersurfaces, and uncover a surprising dichotomy:
the noncompact version of CHP fails in R3 but holds in higher dimensions.
Theorem 1.1. Let M be a connected, smooth n-manifold with compact boundary,

  

Source: Alexander, Stephanie - Department of Mathematics, University of Illinois at Urbana-Champaign

 

Collections: Mathematics