First eigenvalues of geometric operators under the Ricci flow
HTML articles powered by AMS MathViewer
- by Xiaodong Cao PDF
- Proc. Amer. Math. Soc. 136 (2008), 4075-4078 Request permission
Abstract:
In this paper, we prove that the first eigenvalues of $-\Delta + cR$ ($c\geq \frac 14$) are nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized Ricci flow for the cases $c= 1/4$ and $r\le 0$.References
- Xiaodong Cao, Eigenvalues of $(-\Delta +\frac R2)$ on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435–441. MR 2262792, DOI 10.1007/s00208-006-0043-5
- Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007. Geometric aspects. MR 2302600, DOI 10.1090/surv/135
- Shu-Cheng Chang and Peng Lu, Evolution of Yamabe constant under Ricci flow, Ann. Global Anal. Geom. 31 (2007), no. 2, 147–153. MR 2326418, DOI 10.1007/s10455-006-9041-9
- Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, DOI 10.1090/conm/071/954419
- Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
- Thomas Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301–307. MR 1249376, DOI 10.1016/0926-2245(93)90008-O
- Bruce Kleiner and John Lott, Notes on Perelman’s papers, 2006.
- Jun-Fang Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338 (2007), no. 4, 927–946. MR 2317755, DOI 10.1007/s00208-007-0098-y
- —, Monotonicity formulas under rescaled Ricci flow.
- Li Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Global Anal. Geom. 29 (2006), no. 3, 287–292. MR 2248073, DOI 10.1007/s10455-006-9018-8
- T. Oliynyk, V. Suneeta, and E. Woolgar, Irreversibility of world-sheet renormalization group flow, Phys. Lett. B 610 (2005), no. 1-2, 115–121. MR 2118230, DOI 10.1016/j.physletb.2005.01.077
- Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002.
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Additional Information
- Xiaodong Cao
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
- MR Author ID: 775164
- Email: cao@math.cornell.edu
- Received by editor(s): October 5, 2007
- Published electronically: June 2, 2008
- Additional Notes: This research was partially supported by an MSRI postdoctoral fellowship
- Communicated by: Chuu-Lian Terng
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 4075-4078
- MSC (2000): Primary 58C40; Secondary 53C44
- DOI: https://doi.org/10.1090/S0002-9939-08-09533-6
- MathSciNet review: 2425749