Isometric Deformations of Compact Hypersurfaces

Abstract

In 1955 N. Kuiper and J. Nash proved that given a C ^∞ embeddingF of a C ^∞ Riemannian n -manifold (M,g) in E ⁿ⁺¹ which is short in the sense that the metric induced by F is less thang, there is a C ¹ isometric embedding which is arbitrarily C ⁰-close to F. We extend the Nash--Kuiper result for compact M to the case of deformations. In other words, we prove that given a continuous family of short C ^∞ embeddings $F(s):M \to E^{n + 1}$ ($s \in [0,1]$) of a compact Riemannian n-manifold M , there is an isometric deformation through C ¹ embeddings which is C ⁰ -close to F. With more assumptions which are typically met in practice, this result is shown to hold in the more difficult case where F(s) is short for s>0 andF(0) is isometric. We use this to prove that if a C ^∞ convex hypersurface is sufficiently close to being planar in an average sense (e.g. an oblate spheroid in E ³ with axis ratio more than $\sqrt {8/3}$, then it admits an isometric deformation which increases the enclosed volume.