New monotonicity formulas for Ricci curvature and applications; I
Original manuscript November 21, 2011
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Springer-Verlag
2013
|
| Online Access: | http://hdl.handle.net/1721.1/79912 https://orcid.org/0000-0001-6208-384X |
| _version_ | 1826203870065328128 |
|---|---|
| author | Colding, Tobias |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias |
| author_sort | Colding, Tobias |
| collection | MIT |
| description | Original manuscript November 21, 2011 |
| first_indexed | 2024-09-23T12:44:35Z |
| format | Article |
| id | mit-1721.1/79912 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:44:35Z |
| publishDate | 2013 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/799122022-10-01T10:51:25Z New monotonicity formulas for Ricci curvature and applications; I New monotonicity formulas for Ricci curvature and applications. I Colding, Tobias Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Original manuscript November 21, 2011 We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these. Moreover, we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the Green function that we prove. This is parallel to the fact that Perelman’s monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li–Yau. In [CM4] one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross-sections of Einstein manifolds. Finally, there are obvious parallelisms between our monotonicity and the positive mass theorem of Schoen–Yau and Witten. National Science Foundation (U.S.) (Grant DMS-11040934) National Science Foundation (U.S.). Focused Research Group (Grant DMS 0854774) National Science Foundation (U.S.) (Grant 0932078) 2013-08-22T13:33:10Z 2013-08-22T13:33:10Z 2012-12 2011-11 Article http://purl.org/eprint/type/JournalArticle 0001-5962 1871-2509 http://hdl.handle.net/1721.1/79912 Colding, Tobias Holck. “New monotonicity formulas for Ricci curvature and applications. I.” Acta Mathematica 209, no. 2 (December 6, 2012): 229-263. https://orcid.org/0000-0001-6208-384X en_US http://dx.doi.org/10.1007/s11511-012-0086-2 Acta Mathematica Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Springer-Verlag arXiv |
| spellingShingle | Colding, Tobias New monotonicity formulas for Ricci curvature and applications; I |
| title | New monotonicity formulas for Ricci curvature and applications; I |
| title_full | New monotonicity formulas for Ricci curvature and applications; I |
| title_fullStr | New monotonicity formulas for Ricci curvature and applications; I |
| title_full_unstemmed | New monotonicity formulas for Ricci curvature and applications; I |
| title_short | New monotonicity formulas for Ricci curvature and applications; I |
| title_sort | new monotonicity formulas for ricci curvature and applications i |
| url | http://hdl.handle.net/1721.1/79912 https://orcid.org/0000-0001-6208-384X |
| work_keys_str_mv | AT coldingtobias newmonotonicityformulasforriccicurvatureandapplicationsi |