Perelman’s \(\lambda \)-Functional on Manifolds with Conical Singularities


In this paper, we prove that on a compact manifold with isolated conical singularity, the spectrum of the Schrödinger operator \(-4\Delta +R\) consists of discrete eigenvalues with finite multiplicities, if the scalar curvature R satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior for eigenfunctions near the singularity. As a consequence of these spectral properties, we extend the theory of the Perelman’s \(\lambda \)-functional on smooth compact manifolds to compact manifolds with isolated conical singularities.

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  1. 1.

    Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. vol. XXXIX, 661–693 (1986)

  2. 2.

    Bateman, H.: Higher Transcendental Functions, vol. I. McGraw-Hill Book Company, INC, New York (1953)

    Google Scholar 

  3. 3.

    Besse, A.L.: Einstein Manifolds. Springer, New York (1987)

    Google Scholar 

  4. 4.

    Botvinnik, B., Preston, B.: Conformal Laplacian and Conical Singularities. In: Proceeding of the School on High-Dimensional Manifold Topology, ICTP, Trieste, Italy, World Scientific (2003). arXiv:math/0201058v2 [math.DG] (2002)

  5. 5.

    Brüning, J., Seeley, R.: The resolvent expansion for second order regular singular operators. J. Funct. Anal. 73, 369–429 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Am. Math. Soc. 5, 235–262 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helvetici 51, 43–55 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Choquet-Bruhat, Y., Christodoulou, D.: Elliptic systems in \(H_{s,\delta }\) spaces on manifolds which are Euclidean at infinity. Acta Math. 146, 129–150 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Chen, X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds I, II and III. J. Am. Math. Soc. 28, 183–278 (2015)

    Article  MATH  Google Scholar 

  10. 10.

    Chavel, I.: Eigenvalues in Riemmanian Geometry. Academic Press, Inc., Orlando (1984)

    Google Scholar 

  11. 11.

    Cao, H.-D., Hamilton, R.S., Ilmanen, T.: Gaussian densities and stability for some Ricci solitons. arXiv:math.DG/0404165 (2004)

  12. 12.

    Chaljub-Simonn, A., Choquet-Bruhat, Y.: Problèmes elliptiques du second ordre sur une variété euclidienne à l’infini. Ann. Fac. Sci. Toulouse 1, 9–25 (1978)

    Article  MATH  Google Scholar 

  13. 13.

    Cao, H.-D., Zhu, X.-P.: A complete proof of the Poincaré and geometrization conjectures-application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–492 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Cao, H.-D., Zhu, M.: On second variation of Perelman’s Ricci shrinker entropy. Math. Ann. 353, 747–763 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Cheeger, J.: On the spectral geometry of spaces with cone-like singularities. Proc. Natl. Acad. Sci. USA 76(5), 2103–2106 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Dai, X., Wang, X., Wei, G.: On the stability of Riemannian manifold with parallel spinors. Invent. Math. 161(1), 151–176 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Egorov, Y., Kondratiev, V.: On Spectral Theory of Elliptic Operators. Birkhäuser, Basel (1996)

    Google Scholar 

  18. 18.

    Haslhofer, R.: Perelman’s lambda-functional and the stability of Ricci-flat metrics. Calc. Var. PDE 45(3–4), 481–504 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Haslhofer, R., Müller, R.: Dynamical stability and instability of Ricci-flat metrics. Math. Ann. 360(1–2), 547–553 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Kozlov, V., Maz’ya, J., Rossmann, J.: Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs, vol. 52. American Mathematical Society, Providence, RI (1997)

  21. 21.

    Kufner, A.: Weighted Sobolev Spaces. John Wiley & Sons Limited, Chichester (1985)

    Google Scholar 

  22. 22.

    Lockhart, R.: Fredholm properties of a class of elliptic operators on non-compact manifolds. Duke Math. J. 48, 289–312 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Lockhart, R., McOwen, R.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa. 12(3), 409–447 (1985)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Lee, J., Parker, T.: The Yamabe problem. Bull. Am. Math. Soc. 17(1), 833–866 (1987)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mazzeo, R.: Elliptic theory of differential edge operators I. Commun. Part. Diff. Eq. 16(10), 1615–1664 (1991)

  26. 26.

    McOwen, R.: Behavior of the Laplacian on weighted Sobolev spaces. Commun. Pure Appl. Math. 32, 783–795 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Mazzeo, R., Rubinstein, Y.A., Sesum, N.: Ricci flow on surfaces with conic singularities. Anal. PDE 8(4), 839–882 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Nirenberg, L., Walker, H.: The null spaces of elliptic partial differential operators in \({\mathbb{R}}^{n}\). J. Math. Anal. Appl. 42, 271–301 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Ozuch, T.: Perelman’s functionals on cones, construction of type III Ricci flows coming out of cones. arXiv:1707.06102v1 [math. DG] (2017)

  30. 30.

    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002)

  31. 31.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness. Elsevier (Singapore) Pte Ltd, Singapore (1975)

    Google Scholar 

  32. 32.

    Sesum, N.: Linear and dynamical stability of Ricci-flat metrics. Duke Math. J. 133(1), 1–26 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Schoen, R., Yau, S.T.: Lectures on Differential Geometry. International Press of Boston Inc, Boston (1994)

    Google Scholar 

  34. 34.

    Tian, G.: K-stability and Kähler-Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)

    Article  MATH  Google Scholar 

  35. 35.

    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland Publishing Co., Amsterdam (1978)

  36. 36.

    Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Springer, Berlin (2000)

    Google Scholar 

  37. 37.

    Vertman, B.: Ricci flow on singular manifolds. arXiv:1603.06545 (2016)

  38. 38.

    Wang, Y.: An elliptic theory of indicial weights and applications to non-linear geometry problems. arXiv:1702.05864 [math. DG] (2017)

  39. 39.

    Yin, H.: Ricci flow on surfaces with conical singularities. J. Geom. Anal. 20(4), 970–995 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Yin, H.: Ricci flow on surfaces with conical singularities II. arXiv:1305.4355 (2013)

  41. 41.

    Zhang, Q.S.: Extremal of Log Sobolev inequality and W enptropy on noncompact manifolds. J. Funct. Anal. 263, 2051–2101 (2012)

    MathSciNet  Article  MATH  Google Scholar 

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Funding were provided by Directorate for Mathematical and Physical Sciences (Grant No. 1611915) and Simons Foundation.

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Correspondence to Xianzhe Dai.

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Dai, X., Wang, C. Perelman’s \(\lambda \)-Functional on Manifolds with Conical Singularities. J Geom Anal 28, 3657–3689 (2018).

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  • Perelman’s F entropy
  • Perelman’s Lambda functional
  • Manifolds with conical singularity
  • Eigenvalues and eigenfunctions

Mathematics Subject Classification

  • 53C44
  • 58J50