Heat kernel on Ricci shrinkers
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Calculus of Variations
Heat kernel on Ricci shrinkers Yu Li1 · Bing Wang2 Received: 22 July 2019 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong maximum principle for curvature tensors, are essentially improved for Ricci flows induced by Ricci shrinkers. Our results provide many necessary tools to analyze short time singularities of the Ricci flows of general dimension. Mathematics Subject Classification 53C25 · 53E20
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . 3 Cutoff functions, maximum principle and heat kernel 4 Monotonicity of Perelman’s entropy . . . . . . . . . 5 Optimal logarithmic Sobolev constant—part I . . . 6 Optimal logarithmic Sobolev constant—part II . . . 7 Heat kernel estimates . . . . . . . . . . . . . . . . 8 Differential Harnack inequality on Ricci shrinkers . 9 The no-local-collapsing theorems . . . . . . . . . . 10 The pseudolocality theorems . . . . . . . . . . . . . 11 Strong maximum principle for curvature operator . . References . . . . . . . . . . . . . . . . . . . . . . . .
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Communicated by A. Neves. Yu Li is partially supported by research fund from SUNY Stony Brook and Bing Wang is partially supported by NSF Grant DMS-1510401 and research funds from USTC and UW-Madison.
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Yu Li [email protected] Bing Wang [email protected]
1
Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA
2
The Institute of Geometry and Physics, School of Mathematical Sciences, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei 230026, Anhui Province, China 0123456789().: V,-vol
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Y. Li, B. Wang
1 Introduction A Ricci shrinker is a triple (M n , g, f ) of smooth manifold M n , Riemannian metric g and a smooth function f satisfying Rc + Hess f =
1 g. 2
(1)
By a normalization of f , we can assume that
R + |∇ f |2 = f , − n2
e−
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