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PreprintarXiv licencearXiv

Relative volume comparison theorem under Kato type conditions

  • Yuting Wu
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Abstract

Let $(M, g)$ be an n (≥ 3) dimensional, non-collapsed compact Riemannian manifold and Ric- be the negative part of the Ricci curvature and β∈ (2n/(n+2), 2). We prove a relative volume comparison theorem when |Ric-|β is in the Kato class (cf. Definition 1.1), which results from a new integral Laplace comparison theorem in the spirit of \cite{PW} for a suitable conformal metric. This partly addresses an expectation in \cite{TZZZZ}, where the same result was proven when |Ric-|2 is in the class.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026relative,
  title = {{Relative volume comparison theorem under Kato type conditions}},
  author = {Yuting Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30156},
  eprint = {2609.30156},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30156},
}
RIS
TY  - UNPB
TI  - Relative volume comparison theorem under Kato type conditions
AU  - Yuting Wu
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.30156
UR  - https://arxiv.org/abs/2609.30156
AB  - Let $(M, g)$ be an n (≥ 3) dimensional, non-collapsed compact Riemannian manifold and Ric^- be the negative part of the Ricci curvature and β∈ (2n/(n+2), 2). We prove a relative volume comparison theorem when |Ric^-|^β is in the Kato class (cf. Definition 1.1), which results from a new integral Laplace comparison theorem in the spirit of \cite{PW} for a suitable conformal metric. This partly addresses an expectation in \cite{TZZZZ}, where the same result was proven when |Ric^-|^2 is in the class.
ER  -