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FIRST EIGENVALUES OF GEOMETRIC OPERATORS UNDER THE YAMABE FLOW
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 Title & Authors
FIRST EIGENVALUES OF GEOMETRIC OPERATORS UNDER THE YAMABE FLOW
Fang, Shouwen; Yang, Fei;
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 Abstract
Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Yamabe flow. In the paper we derive the evolution for the first eigenvalue of geometric operator under the Yamabe flow, where is the Witten-Laplacian operator, , and R is the scalar curvature with respect to the metric g(t). As a consequence, we construct some monotonic quantities under the Yamabe flow.
 Keywords
eigenvalue;Witten-Laplacian;Yamabe flow;
 Language
English
 Cited by
 References
1.
X. D. Cao, Eigenvalues of ($-{\Delta}+\frac{R}{2}$) on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435-441.

2.
X. D. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4075-4078. crossref(new window)

3.
S. W. Fang, H. F. Xu, and P. Zhu, Evolution and monotonicity of eigenvalues under the Ricci flow, Sci. China Math. 58 (2015), no. 8, 1737-1744. crossref(new window)

4.
S. W. Fang, F. Yang, and P. Zhu, Eigenvalues of geometric operators related to the Witten-Laplacian under the Ricci flow, preprint.

5.
H. X. Guo, R. Philipowski, and A. Thalmaier, Entropy and lowest eigenvalue on evolving manifolds, Pacific J. Math. 264 (2013), no. 1, 61-81. crossref(new window)

6.
B. Kleiner and J. Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008), no. 5, 2587-2858. crossref(new window)

7.
J. F. Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338 (2007), no. 4, 927-946. crossref(new window)

8.
X. D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. (9) 84 (2005), no. 10, 1295-1361. crossref(new window)

9.
J. Ling, A comparison theorem and a sharp bound via the Ricci flow, http://arxiv.org/abs/0710.2574.

10.
J. Ling, A class of monotonic quantities along the Ricci flow, http://arxiv.org/abs/0710.4291.

11.
L. Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Global Anal. Geom. 29 (2006), no. 3, 287-292. crossref(new window)

12.
P. Topping, Lectures on the Ricci Flow, Vol. 325, Cambridge University Press, 2006.

13.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arxiv.org/abs/math/0211159.

14.
C. Y. Xia and H. W. Xu, Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann. Global Anal. Geom. 45 (2014), no. 3, 155-166. crossref(new window)

15.
L. Zhao, The first eigenvalue of the Laplace operator under Yamabe flow, Math. Appl. 24 (2011), no. 2, 274-278.

16.
L. Zhao, The first eigenvalue of the p-Laplace operator under powers of the mth mean curvature flow, Results Math. 63 (2013), no. 3-4, 937-948. crossref(new window)

17.
L. Zhao, The first eigenvalue of the p-Laplace operator under powers of mean curvature flow, Math. Methods Appl. Sci. 37 (2014), no. 5, 744-751. crossref(new window)