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Logarithmic Sobolev and exponential inequalities for submanifolds in Cartan--Hadamard manifolds

  • Hongyu Chen
  • Li Ma
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Abstract

We establish a logarithmic Sobolev inequality for complete C1,1 immersed submanifolds of dimension m≥2 in Cartan--Hadamard manifolds with sectional curvature at most −b2. The inequality retains the Euclidean sharp constant after subtraction of (m−1)2b2 from the Fisher and mean-curvature energy; this coefficient is maximal for a lower bound uniform over b≥0. For immersed surfaces, we prove both a finite-domain critical exponential inequality and a global Ruf-type inequality with a positive L2 term. Both have potential |H|2−b2, exponent 16π and an absolute multiplicative constant, and follow from one geometric heat-kernel comparison. The exponent and curvature coefficient are optimal. We classify Gaussian equality in the logarithmic inequality and geometric equality in the higher-dimensional Nash consequence. All inequality constants are independent of the codimension.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026logarithmic,
  title = {{Logarithmic Sobolev and exponential inequalities for submanifolds in Cartan--Hadamard manifolds}},
  author = {Hongyu Chen and Li Ma},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01610},
  eprint = {2611.01610},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01610},
}
RIS
TY  - UNPB
TI  - Logarithmic Sobolev and exponential inequalities for submanifolds in Cartan--Hadamard manifolds
AU  - Hongyu Chen
AU  - Li Ma
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01610
UR  - https://arxiv.org/abs/2611.01610
AB  - We establish a logarithmic Sobolev inequality for complete C^1,1 immersed submanifolds of dimension m≥2 in Cartan--Hadamard manifolds with sectional curvature at most −b^2. The inequality retains the Euclidean sharp constant after subtraction of (m−1)^2b^2 from the Fisher and mean-curvature energy; this coefficient is maximal for a lower bound uniform over b≥0. For immersed surfaces, we prove both a finite-domain critical exponential inequality and a global Ruf-type inequality with a positive L^2 term. Both have potential |H|^2−b^2, exponent 16π and an absolute multiplicative constant, and follow from one geometric heat-kernel comparison. The exponent and curvature coefficient are optimal. We classify Gaussian equality in the logarithmic inequality and geometric equality in the higher-dimensional Nash consequence. All inequality constants are independent of the codimension.
ER  -