Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem
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.
Keywords
Cite
@article{arxiv.1204.1408,
title = {Deformation of Hypersurfaces Preserving the Moebius Metric and a Reduction Theorem},
author = {Tongzhu Li and Xiang Ma and Changping Wang},
journal= {arXiv preprint arXiv:1204.1408},
year = {2014}
}
Comments
51 pages. A mistake in the proof to Theorem 9.2 has been fixed. Accepted by Adv. in Math