Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem

TL;DR

This paper investigates the rigidity and deformation of hypersurfaces in high-dimensional Euclidean spaces that preserve the Moebius metric, providing classification results and a reduction theorem for constructing non-trivial examples.

Contribution

It establishes conditions for Moebius rigidity, classifies deformable hypersurfaces, and introduces a Reduction Theorem for constructing and understanding hypersurface deformations.

Findings

Hypersurfaces with principal curvature multiplicity less than n-2 are Moebius rigid.

Complete classification of deformable hypersurfaces preserving the Moebius metric.

A Reduction Theorem characterizing classical hypersurface constructions.

Abstract

A hypersurface without umbilics in the n+1 dimensional Euclidean space is known to be determined by the Moebius metric and the Moebius second fundamental form up to a Moebius transformation when n>2. In this paper we consider Moebius rigidity for hypersurfaces and deformations of a hypersurface preserving the Moebius metric in the high dimensional case n>3. When the highest multiplicity of principal curvatures is less than n-2, the hypersurface is Moebius rigid. Deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a Reduction Theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future.