The Intrinsic Structure of High-Dimensional Data According to the Uniqueness of Constant Mean Curvature Hypersurfaces

Abstract

In this paper, we study the intrinsic structures of high-dimensional data sets for analyzing their geometrical properties, where the core message of the high-dimensional data is hiding on some nonlinear manifolds. Using the manifold learning technique with a particular focus on the mean curvature, we develop new methods to investigate the uniqueness of constant mean curvature spacelike hypersurfaces in the Lorentzian warped product manifolds. Furthermore, we extend the uniqueness of stochastically complete hypersurfaces using the weak maximum principle. For the more general cases, we propose some non-existence results and a priori estimates for the constant higher-order mean curvature spacelike hypersurface.

Keywords

• intrinsic geometry structure
• Laplacian operator
• weak maximum principle
• uniqueness
• spacelike slice
• spacelike hypersurface