Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces

By Nicola Garofalo, Andrea Mondino. 2013

76 citations recorded by the source corpus.

Abstract Metric measure spaces satisfying the reduced curvature-dimension condition CD ∗ ( K , N ) and where the heat flow is linear are called RCD ∗ ( K , N ) -spaces. This class of non smooth spaces contains Gromov–Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below by K and dimension bounded above by N . We prove that in RCD ∗ ( K , N ) -spaces the following properties of the heat flow hold true: a Li–Yau type inequality, a Bakry–Qian inequality, the Harnack inequality.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2da09bc9a91400a4b10056b55f8b84b50a3967ad

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In Applied Mathematics

1. Andrea Mondino
2. Fabio Cavalletti
3. Franccois Bolley
4. Meng Zhu
5. Ivan Gentil
6. D. Bakry
7. Nicola Gigli
8. Qi S. Zhang

In Astronomy and Astrophysics

1. Fabio Cavalletti
2. Andrea Mondino
3. Meng Zhu
4. Qi S. Zhang
5. Nicola Gigli
6. Luigi Ambrosio
7. Emanuel Milman
8. Yu Kitabeppu

In Geometry and Topology

1. Andrea Mondino
2. Fabio Cavalletti
3. Meng Zhu
4. Franccois Bolley
5. Ivan Gentil
6. Qi S. Zhang
7. D. Bakry
8. A. Naber

Citations in the corpus

• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “While a great deal of progress has been made in this latter general framework from both the analytic, geometric and structural points of view, see for instance [2, 3, 4, 5, 6, 7, 9, 14, 15, 18, 27, 33, 35, 37, 36, 43, 50, 59, 63, 66], the isoperimetric problem has remained elusive.”
• The Li-Yau inequality and applications under a curvature-dimension condition (2014, background): “Let us mention in particular the most recent [15, 19, 4, 12, 9, 14, 7, 13].”
• Li–Yau gradient bounds on compact manifolds under nearly optimal curvature conditions (2015, background): “the Li-Yau bound for weighted manifolds by an ordinary differential inequality involving the entropy and energy of the backward heat equation. For most recent development, see the papers [CTZ], [Dav], [GM], [LX], [QZZ], [Wan], [WanJ] and the latest [BBG], [ZZ] and references therein. In all of these ”