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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    Henceforth, the term \({ monotonicity}\) will always mean non-decreasing in time.

References

  1. 1.

    Besse, A.: Einstein Manifolds, Classics in Mathematics. Springer, New York (2008)

    Google Scholar 

  2. 2.

    Cao, X.: Eigevalues of \(-\Delta +\dfrac{R}{2}\) on manifolds with nonnegative curvature operator. Math. Ann. 337, 435–441 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cao, X.: First eigenvalues of geometric operators under the Ricci flow. Proc. Am. Math. Soc. 136, 4075–4078 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cao, X., Hou, S., Ling: Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354, 451–463 (2012)

    MathSciNet  Article  Google Scholar 

  5. 5.

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

  6. 6.

    Di Cerbo, L.F.: Eigenvalues of the Laplacian under the Ricci flow. Rendiconti di Matematica, Serie VII 27, 183–195 (2007)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Fang, S., Xu, H., Zhu, P.: Evolution and monotonicity of eigenvalues under the Ricci flow. Sci. China Math. 58, 1737–1744 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Fang, S., Zhao, L., Zhu, P.: Estimates and monotonicity of the first eigenvalues under the ricci flow on closed surfaces. Commun. Math. Stat. 4, 217–228 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Fang, S., Yang, F.: First eigenvalues of geometric operators under the Yamabe flow. Bull. Korean Math. Soc. 53(4), 1113–1122 (2016)

    MathSciNet  Article  Google Scholar 

  10. 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. 11.

    Li, J.: Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338(4), 927–946 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kriegl, A., Michor, P.W.: Differentiable perturbation of unbounded operators. Math. Ann. 327(1), 191–201 (2003)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Ma, L.: Eigenvalue monotonicity for the Ricci–Hamilton flow. Ann. Global Anal. Geom. 29(3), 287–292 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

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

  16. 16.

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

  17. 17.

    Zeng, F., He, Q., Chen, B.: Monotonicity of eigenvalues of geometric operators along the Ricci–Bourguignon flow. Pac. J. Math. 296(1), 1–20 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We would like to thank Jose N. V. Gomes and Marcus Marrocos for stimulating discussions. The first named author sincerely acknowledges the hospitality of Henrique Sá Earp and IMECC at University of Campinas as well as the generous financial support by FAPESP in terms of the Grant 2016/06395-5.

Author information

Affiliations

Authors

Corresponding author

Correspondence to D. M. Tsonev.

Appendix: An Alternative Proof of the Evolution Formula

Appendix: An Alternative Proof of the Evolution Formula

As promised earlier in Sect. 3, we shall now give an alternative proof of the evolution formula for the eigenvalues of the operator \(-\mathbb {L}\) evolving with the flow (1.1). This proof conceptually differs from the one we have already presented, for it makes use of the notion of the \(\eta \)-divergence and some of its properties. Given a symmetric (0, r)-tensor S on M we define the \(\eta \)-divergence to be the \((0,r-1)\)-tensor

$$\begin{aligned} \text {div}_{\eta }S=\text {div} S-\text {d}\eta \circ S, \end{aligned}$$

where \(\text {div}S\) is the usual divergence of S and \(\eta \) is the drifting function. The identity \(\text {div}_{\eta }(\nabla f)=L(f)\) is effortlessly checked. The \(\eta \)-divergence enjoys the property \(\text {div}(e^{-\eta }X)=e^{-\eta }\text {div}_{\eta }X\) for any smooth function f and a vector field X on the manifold M. It is by its virtue that we have a natural extension of the divergence theorem for \(\text {div}_{\eta }\). Indeed, given a vector field X on M and the weighted measure \(\text {d}\mu =e^{-\eta }\text {d}\partial M\) on the boundary \(\partial M\), the divergence theorem takes the form

$$\begin{aligned} \int _M\text {div}_{\eta }X \text {d}m=\int _{\partial M}g(X,\nu )\text {d}\mu , \end{aligned}$$

where \(\nu \) is the unit outward normal vector. We shall also need in what follows the property

$$\begin{aligned} \text {div}_{\eta }(fX)=f\text {div}_{\eta }X+g(\nabla f, X). \end{aligned}$$

Taking any function \(f\in C_{0}^{\infty }(M)\) and working in coordinates one can straightforwardly verify the identity

$$\begin{aligned} \frac{\text {d}}{\text {d}t}\bigg (\mathbb {L}f\bigg )=\mathbb {L}'f+\mathbb {L}f'. \end{aligned}$$
(8.1)

Moreover, the following formula holds true (see [10])

$$\begin{aligned} \mathbb {L}'f=\bigg \langle \frac{1}{2}\text {d}h-\text {div}_{\eta }{\mathcal {H}},\text {d}f\bigg \rangle -\langle {\mathcal {H}},\nabla ^2f\rangle +cR'f. \end{aligned}$$
(8.2)

Now, with the aforementioned preparatory remarks in mind we can give the second proof of the evolution formula (3.1).

Proof

To begin with, observe that by differentiating in t the obvious identity

$$\begin{aligned} -u(t)\mathbb {L}u(t)=\lambda (t)u^2(t), \end{aligned}$$

we easily obtain

$$\begin{aligned} u'\mathbb {L}u-u\mathbb {L}u'=\lambda 'u^{2}+u\mathbb {L}'u, \end{aligned}$$

which reduces to

$$\begin{aligned} u'Lu-uLu'= \lambda ' u^{2}+u\mathbb {L}'u. \end{aligned}$$

By dint of formula (2.8) we readily perceive that the left hand side vanishes after the integration and therefore the following simple formula holds true

$$\begin{aligned} \lambda '=-\int _{M}u\mathbb {L}'u\,\text {d}m. \end{aligned}$$
(8.3)

It is now evident that we only need to compute the integral on the right hand side. One can calculate this integral in two different ways. The obvious one is to use the variation formula for the operator \(\mathbb {L}\). Namely, plugging (8.2) in (8.3) and using the properties of \(\text {div}_{\eta }\) as well as the divergence theorem one will eventually arrive at the desired formula. Leaving this computation for the reader to check we shall give another way of computing the integral in (8.3). We observe first that the definition of \(\mathbb {L}\) and the integration by part formula yield

$$\begin{aligned} \int _{M}u\mathbb {L}u\,\text {d}m=-\int _{M}\langle \text {d}u,\text {d}u\rangle \text {d}m+c\int _{M}Ru^{2}\text {d}m. \end{aligned}$$
(8.4)

Evidently, we only need to differentiate the latter formula and simplify to formula (3.1). Using the identity (8.1) we can rewrite the left hand side integral of (8.4) as

$$\begin{aligned} \dfrac{\text {d}}{\text {d}t}\left( \int _{M}u\mathbb {L}u\,\text {d}m\right) =\int _{M}u'(\mathbb {L}u)\text {d}m+\int _{M}u\mathbb {L}'u\,\text {d}m+\int _{M}u\mathbb {L}u'\,\text {d}m+\frac{1}{2}\int _{M}hu\mathbb {L}u\,\text {d}m. \end{aligned}$$

Differentiation of the right hand side of (8.4) gives

$$\begin{aligned} \dfrac{\text {d}}{\text {d}t}\left( \int _{M}u\mathbb {L}u\,\text {d}m\right)= & {} -2\int _{M}\langle \text {d}u,\text {d}u'\rangle \text {d}m+\int _{M}{\mathcal {H}}(\nabla u,\nabla u)\text {d}m-\frac{1}{2}\int _{M}h|\nabla u|^{2}\text {d}m\\&+\,c\int _{M}R'u^{2}\text {d}m+c\int _{M}R(u^{2})'\text {d}m+\frac{c}{2}\int _{M}hRu^{2}\text {d}m. \end{aligned}$$

It is also easily seen that the integration by parts formula implies

$$\begin{aligned} \int _{M}u'\mathbb {L}u\,\text {d}m+\int _{M}u\mathbb {L}u'\text {d}m= & {} \int _{M}u'Lu\,\text {d}m+\int _{M}uLu'\,\text {d}m+c\int _{M}R(u^{2})'\text {d}m\\= & {} -\,2\int _{M}\langle \text {d}u,\text {d}u'\rangle \text {d}m+c\int _{M}R(u^{2})'\text {d}m. \end{aligned}$$

Thus, the latter three formulae justify the validity of

$$\begin{aligned} \int _{M}u\mathbb {L'}u\,\text {d}m= & {} \int _{M}{\mathcal {H}}(\nabla u,\nabla u)\text {d}m-\frac{1}{2}\int _{M}h\langle \text {d}u,\text {d}u\rangle \text {d}m\nonumber \\&+\,c\int _{M}R'u^{2}\text {d}m+\frac{c}{2}\int _{M}Rhu^{2}\text {d}m-\frac{1}{2}\int _{M}hu\mathbb {L}u\,\text {d}m. \end{aligned}$$
(8.5)

We shall have to make a little detour at this point. Using the properties of \(\text {div}_{\eta }\) we readily compute

$$\begin{aligned} \text {div}_{\eta }(hu\nabla u)= & {} huLu+g(\nabla hu,\nabla u) \\= & {} huLu+h\langle \text {d}u, \text {d}u\rangle +u\langle \text {d}h, \text {d}u\rangle \\= & {} hu(\mathbb {L}u-cRu)+h\langle \text {d}u, \text {d}u\rangle +u\langle \text {d}h, \text {d}u\rangle \\= & {} hu\mathbb {L}u-cRhu^{2}+h\langle \text {d}u, \text {d}u\rangle +u\langle \text {d}h, \text {d}u\rangle . \end{aligned}$$

Now, as we are on a closed manifold, \(\int _M\text {div}_{\eta }(hu\nabla u) \text {d}m\) vanishes and we obtain

$$\begin{aligned} \int _{M}h\langle \text {d}u, \text {d}u\rangle \text {d}m=-\int _{M}u\langle \text {d}h, \text {d}u\rangle \text {d}m-\int _{M}hu\mathbb {L}u\,\text {d}m+c\int _{M}Rhu^{2}\text {d}m. \end{aligned}$$

It follows that relation (8.5) can be expressed as

$$\begin{aligned} \int _{M}u\mathbb {L}'u\,\text {d}m=\int _{M}{\mathcal {H}}(\nabla u,\nabla u)\text {d}m+\frac{1}{2}\int _{M}u\langle \text {d}h, \text {d}u\rangle \text {d}m+c\int _{M}R'u^{2}\text {d}m. \end{aligned}$$

Recall now that the Witten-Laplacian can be defined as \(\text {div}_{\eta }(\nabla f)=L(f)\). This implies the identity

$$\begin{aligned} \frac{u}{2}\langle \text {d}h, \text {d}u\rangle =\frac{1}{4}\text {div}_{\eta }(h\nabla u^{2})-\frac{1}{4}hL(u^{2}), \end{aligned}$$

which, by dint of the divergence theorem, can be rewritten in the following integral form

$$\begin{aligned} \int _M\frac{u}{2}\langle \text {d}h, \text {d}u\rangle \text {d}m=-\int _M\frac{1}{4}hL(u^{2}) \text {d}m. \end{aligned}$$

We can then write

$$\begin{aligned} \int _{M}u\mathbb {L}'u\,\text {d}m=\int _{M}\left( -\frac{h}{4}L(u^{2})+{\mathcal {H}}(\nabla u,\nabla u)+cR'u^{2}\right) \text {d}m, \end{aligned}$$

which completes the proof. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tsonev, D.M., Mesquita, R.R. On the Spectra of a Family of Geometric Operators Evolving with Geometric Flows. Commun. Math. Stat. (2021). https://doi.org/10.1007/s40304-020-00215-6

Download citation

Keywords

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

Mathematics Subject Classification

  • 53E20
  • 53E99
  • 58C40