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

Junhong Dong, Qiong Li () and Ximin Liu
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Junhong Dong: School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519000, China
Qiong Li: Department of Statistics and Data Science, BNU-HKBU United International College, Zhuhai 519087, China
Ximin Liu: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Mathematics, 2022, vol. 10, issue 20, 1-18

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; constant mean curvature (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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