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Heat kernel on Ricci shrinker metric measure spaces
Abstract
As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function f admits a heat kernel Hf under the weighted volume measure e−fdv. In this paper, we study Hf systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between Hf and the spacetime heat kernel $H(x,t;y,s)$ under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that |∇ R|=o(f3/2) implies R=o(f).
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BibTeX
@misc{wang2026heat,
title = {{Heat kernel on Ricci shrinker metric measure spaces}},
author = {Bing Wang and Jie Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18594},
eprint = {2609.18594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18594},
}
RIS
TY - UNPB
TI - Heat kernel on Ricci shrinker metric measure spaces
AU - Bing Wang
AU - Jie Wang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18594
UR - https://arxiv.org/abs/2609.18594
AB - As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function f admits a heat kernel H_f under the weighted volume measure e^−fdv. In this paper, we study H_f systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between H_f and the spacetime heat kernel $H(x,t;y,s)$ under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that |∇ R|=o(f^3/2) implies R=o(f).
ER -