On the Spectra of a Family of Geometric Operators Evolving with Geometric Flows

Abstract

In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain family of geometric operators under some geometric flows. In an attempt to understand the arising similarities we formulate two conjectures on the monotonicity of the eigenvalues of Schrödinger operators under geometric flows. We also pose three questions which we consider to be of a general interest.

Keywords

Witten-Laplacian / Eigenvalues / Monotonicity of eigenvalues / Ricci flow / Ricci–Bourguignon flow / Yamabe flow / Bochner formula / Reilly formula

References

[1] Besse A. Einstein Manifolds, Classics in Mathematics. 2008 New York: Springer

[2] Cao X. Eigevalues of $-\Delta +\dfrac{R}{2}$ on manifolds with nonnegative curvature operator. Math. Ann.. 2007, 337 435-441

[3] Cao X. First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc.. 2008, 136 4075-4078

[4] Cao X, Hou S, Ling. Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann.. 2012, 354 451-463

[5] Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, Graduate Studies in Mathematics, Vol. 77. American Mathematical Society (2006)

[6] Di Cerbo LF. Eigenvalues of the Laplacian under the Ricci flow. Rendiconti di Matematica, Serie VII. 2007, 27 183-195

[7] Fang S, Xu H, Zhu P. Evolution and monotonicity of eigenvalues under the Ricci flow. Sci. China Math.. 2015, 58 1737-1744

[8] Fang S, Zhao L, Zhu P. Estimates and monotonicity of the first eigenvalues under the ricci flow on closed surfaces. Commun. Math. Stat.. 2016, 4 217-228

[9] Fang S, Yang F. First eigenvalues of geometric operators under the Yamabe flow. Bull. Korean Math. Soc.. 2016, 53 4 1113-1122

[10] Gomes, J.N., Marrocos, M.A.M., Mesquita, R.R.: Hadamard Type Variation Formulas for the Eigenvalues of the $\eta $-Laplacian and Applications. arXiv:1510.07076 [math.DG] (2015)

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

[12] Kriegl A, Michor PW. Differentiable perturbation of unbounded operators. Math. Ann.. 2003, 327 1 191-201

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

[14] Ma L, Du S-H. Extension of Reilly formula with applications to drifting laplacians. Comptes Rendus Mathematique. 2010, 348 21–22 1203-1206

[15] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159

[16] Topping, P.: Lectures on the Ricci Flow, London Mathematical Society Lecture Notes (Vol. 325). Cambridge University Press, Cambridge (2006)

[17] Zeng F, He Q, Chen B. Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow. Pac. J. Math.. 2018, 296 1 1-20

Funding

FAPESP