Estimates and Monotonicity of the First Eigenvalues Under the Ricci Flow on Closed Surfaces

Abstract

In the paper we first derive the evolution equation for eigenvalues of geometric operator $-\Delta _{\phi }+cR$ under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M, where $\Delta _{\phi }$ is the Witten–Laplacian operator, $\phi \in C^{\infty }(M)$, and R is the scalar curvature. We then prove that the first eigenvalue of the geometric operator is nondecreasing along the Ricci flow on closed surfaces with certain curvature conditions when $0<c\le \frac{1}{2}$. As an application, we obtain some monotonicity formulae and estimates for the first eigenvalue on closed surfaces.

Keywords

First eigenvalue / Witten–Laplacian / Ricci flow

References

[1] Cao XD. Eigenvalues of $(-\Delta + \frac{R}{2})$ on manifolds with nonnegative curvature operator. Math. Ann.. 2007, 337 2 435-441

[2] Cao XD. First eigenvalues geometric operators under the Ricci flow. Proc. Am. Math. Soc.. 2008, 136 11 4075-4078

[3] Cao XD, Hou SB, Ling J. Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann.. 2012, 354 2 451-463

[4] Fang, S.W., Xu, H.F., Zhu, P.: Evolution and monotonicity of eigenvalues under the Ricci flow, Sci. China Math. 58. doi:10.1007/s11425-014-4943-7 (2015)

[5] Fang, S.W., Yang, F.: First eigenvalues of geometric operators under the Yamabe flow, preprint

[6] Fang, S.W., Yang, F., Zhu, P.: Eigenvalues of geometric operators related to the Witten–Laplacian under the Ricci flow, preprint

[7] Guo HX, Philipowski R, Thalmaier A. Entropy and lowest eigenvalue on evolving manifolds. Pacific J. Math.. 2013, 264 1 61-82

[8] Li JF. Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann.. 2007, 338 4 927-946

[9] Li XD. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl.. 2005, 54 1295-1361

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

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

[12] Ma L. Eigenvalue monotonicity for the Ricci-Hamilton flow. Ann. Global Anal. Geom.. 2006, 29 3 287-292

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

[14] Xia CY, Xu HW. Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds. Ann. Global Anal. Geom.. 2014, 45 3 155-166

[15] Zhao L. The first eigenvalue of the Laplace operator under Yamabe flow. Math. Appl.. 2011, 24 2 274-278

[16] Zhao L. The first eigenvalue of the $p$-Laplace operator under powers of the mth mean curvature flow. Results. Math.. 2013, 63 937-948

[17] Zhao L. The first eigenvalue of the $p$-Laplace operator under powers of mean curvature flow. Math. Method. Appl. Sci.. 2014, 37 5 744-751

Funding

NSFC(11371310, 11401514)

the University Science Research Project of Jiangsu Province(13KJB110029)

National Natural Science Foundation of China(11471145)

the Fundamental Research Funds for the Central Universities (NS2014076)

the Natural Science Foundation of Jiangsu Province(BK20140804)

Qing Lan Project