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First eigenvalues of geometric operators under the Ricci flow

Author(s): Xiaodong Cao
Journal: Proc. Amer. Math. Soc. 136 (2008), 4075-4078.
MSC (2000): Primary 58C40; Secondary 53C44
Posted: June 2, 2008
MathSciNet review: 2425749
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Abstract | References | Similar articles | Additional information

Abstract: In this paper, we prove that the first eigenvalues of $ -\Delta + cR$ ( $ c\geq \frac14$) 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:

[Cao07]
Xiaodong Cao, Eigenvalues of $ (-\Delta+\frac R2)$ on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435-441. MR 2262792 (2007g:53071)

[CCG$ ^+$07]
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, Geometric aspects. Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007. MR 2302600

[CL07]
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

[Ham88]
Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Amer. Math. Soc., Providence, RI, 1988, pp. 237-262. MR 89i:53029

[Ham95]
-, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Internat. Press, Cambridge, MA, 1995, pp. 7-136. MR 97e:53075

[Ive93]
Thomas Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301-307. MR 94j:53048

[KL06]
Bruce Kleiner and John Lott, Notes on Perelman's papers, 2006.

[Li07a]
Junfang Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338 (2007), no. 4, 927-946. MR 2317755 (2008c:53068)

[Li07b]
-, Monotonicity formulas under rescaled Ricci flow.

[Ma06]
Li Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Global Anal. Geom. 29 (2006), no. 3, 287-292. MR 2248073 (2007c:53086)

[OSW05]
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 (2005h:81232)

[Per02]
Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002.

[RS78]
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analy- sis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. MR 0493421 (58:12429c)

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Additional Information:

Xiaodong Cao
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201
Email: cao@math.cornell.edu

DOI: 10.1090/S0002-9939-08-09533-6
PII: S 0002-9939(08)09533-6
Received by editor(s): October 5, 2007
Posted: June 2, 2008
Additional Notes: This research was partially supported by an MSRI postdoctoral fellowship
Communicated by: Chuu-Lian Terng
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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