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Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions
Author(s) -
Portegies Jacobus W.
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21565
Subject(s) - mathematics , ricci curvature , eigenfunction , riemannian manifold , mathematical analysis , laplace–beltrami operator , heat kernel , bounded function , manifold (fluid mechanics) , laplace operator , heat equation , isometry (riemannian geometry) , dimension (graph theory) , pure mathematics , upper and lower bounds , finite volume method , curvature , geometry , eigenvalues and eigenvectors , p laplacian , mechanical engineering , physics , quantum mechanics , mechanics , engineering , boundary value problem
We show that any closed n ‐dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can be bounded in terms of the dimension, a lower bound on the Ricci curvature, the injectivity radius, and the volume. It follows that the manifold can be embedded by a finite number of eigenfunctions of the Laplace operator. Again, this number only depends on the geometric bounds and the dimension. In addition, both maps can be made arbitrarily close to an isometry. In the appendix, we derive quantitative estimates of the harmonic radius, so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well. © 2016 Wiley Periodicals, Inc.