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Heat kernel on smooth metric measure spaces and applications

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Abstract

We derive a Harnack inequality for positive solutions of the f-heat equation and Gaussian upper and lower bound estimates for the f-heat kernel on complete smooth metric measure spaces with Bakry–Émery Ricci curvature bounded below. Both upper and lower bound estimates are sharp when the Bakry–Émery Ricci curvature is nonnegative. The main argument is the De Giorgi–Nash–Moser theory. As applications, we prove an \(L_f^1\)-Liouville theorem for f-subharmonic functions and an \(L_f^1\)-uniqueness theorem for f-heat equations when f has at most linear growth. We also obtain eigenvalues estimates and f-Green’s function estimates for the f-Laplace operator.

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References

  1. Andrews, B., Ni, L.: Eigenvalue comparison on Bakry–Émery manifolds. Commun. Partial Differ. Equ. 37(11), 2081–2092 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakry, D., Émery, M.: Diffusion hypercontractivitives. In: Séminaire de Probabilités XIX, 1983/1984. Lecture Notes in Math., vol. 1123, pp. 177–206. Springer, Berlin (1985)

  3. Bakry, D., Qian, Z.-M.: Some new results on eigenvectors via dimension, diameter and Ricci curvature. Adv. Math. 155, 98–153 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brighton, K.: A Liouville-type theorem for smooth metric measure spaces. J. Geom. Anal. 23, 562–570 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buser, P.: A note on the isoperimetric constant. Ann. Sci. Ecole Norm. Sup. 15, 213–230 (1982)

    MathSciNet  MATH  Google Scholar 

  6. Cao, H.-D.: Recent progress on Ricci solitons. In: Recent advances in geometric analysis. Adv. Lect. Math. (ALM), vol. 11, pp. 1–38. International Press, Somerville (2010)

  7. Cao, H.-D., Zhou, D.: On complete gradient shrinking Ricci solitons. J. Differ. Geom. 85, 175–186 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Case, J., Shu, Y.-S., Wei, G.: Rigidity of quasi-Einstein metrics. Differ. Geo. Appl. 29, 93–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charalambous, N., Lu, Z.: Heat kernel estimates and the essential spectrum on weighted manifolds. J. Geom. Anal. 25, 536–563 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Charalambous, N., Lu, Z.: The \(L^1\) Liouville property on weighted manifolds. arXiv:1402.6170

  11. Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82, 363–382 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Cheng, X., Zhou, D.: Eigenvalues of the drifted Laplacian on complete metric measure spaces. arXiv:1305.4116

  13. Colbois, B., Soufi, A., Savo, A.: Eigenvalues of the Laplacian on a compact manifold with density. arXiv:1310.1490

  14. Dai, X., Sung, J., Wang, J., Wei, G.: in preparation

  15. Davies, E.B.: Heat kernels and spectral theory. In: Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)

  16. Fernández-López, M., García-Río, E.: Maximum principles and gradient Ricci solitons. J. Differ. Equ. 251, 73–81 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Futaki, A., Li, H., Li, X.-D.: On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons. Ann. Glob. Anal. Geom. 44, 105–114 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grigor’yan, A.: The heat equation on noncompact Riemannian manifolds (Russian). Math. Sb. 182, 55–87 (1991). [translation in Math. USSR Sb. 72, 47–77 (1992)]

  19. Grigor’yan, A.: Heat kernel and analysis on manifolds. In: AMS/IP Studies in Advanced Mathematics, vol. 47. Am. Math. Soc., Providence; International Press, Boston (2009)

  20. Grigor’yan, A.: Heat kernels on weighted manifolds and applications. The ubiquitous heat kernel. In: Contemp. Math., vol. 398, pp. 93–191. Am. Math. Soc., Providence (2006)

  21. Hajłasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sr. I Math. 320, 1211–1215 (1995)

  22. Hamilton, R.: The formation of singularities in the Ricci flow. In: Surveys in Differential Geom., vol. 2, pp. 7-136. International Press, Cambridge (1995)

  23. Hassannezhad, A.: Eigenvalues of perturbed Laplace operators on compact manifolds. Pac. J. Math. 264, 333–354 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Li, P.: Uniqueness of \(L^1\) solutions for the Laplace equation and the heat equation on Riemannian manifolds. J. Differ. Geom. 20, 447–457 (1984)

    MathSciNet  MATH  Google Scholar 

  25. Li, P., Tam, L.-F.: Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set. Ann. Math. 125, 171–207 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, P., Tam, L.-F.: Symmetric Green’s functions on complete manifolds. Am. J. Math. 109, 1129–1154 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, P., Yau, S.-T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156, 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  28. Li, X.-D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pure Appl. 84, 1295–1361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lott, J.: Some geometric properties of the Bakry–Émery–Ricci tensor. Comment. Math. Helv. 78, 865–883 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Malgrange, M.: Existence et approximation des solutions der équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier 6, 271–355 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  32. Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  33. Moser, J.: On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24, 727–740 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  34. Munteanu, O., Sesum, N.: On gradient Ricci solitons. J. Geom. Anal. 23, 539–561 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Munteanu, O., Wang, J.: Smooth metric measure spaces with nonnegative curvature. Commun. Anal. Geom. 19, 451–486 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Munteanu, O., Wang, J.: Analysis of weighted Laplacian and applications to Ricci solitons. Commun. Anal. Geom. 20, 55–94 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Munteanu, O., Wang, J.: Geometry of manifolds with densities. Adv. Math. 259, 269–305 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons. Pac. J. Math. 241, 329–345 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pigola, S., Rimoldi, M., Setti, A.G.: Remarks on non-compact gradient Ricci solitons. Math. Z. 268, 777–790 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Qian, Z.: Estimates for weighted volumes and applications. Q. J. Math. Oxf. Ser. 48, 235–242 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 1992(2), 27–38 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  42. Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differ. Geom. 36, 417–450 (1992)

    MathSciNet  MATH  Google Scholar 

  43. Saloff-Coste, L.: Aspects of Sobolev-type inequalities. In: London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002)

  44. Song, B.-Y., Wei, G., Wu, G.-Q.: Monotonicity formulas for Bakry–Émery Ricci curvature. arXiv:1307.0477

  45. Varopoulos, N.: The Poisson kernel on positively curved manifolds. J. Funct. Anal. 44, 359–380 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wei, G., Wylie, W.: Comparison geometry for the Bakry–Émery Ricci tensor. J. Differ. Geom. 83, 377–405 (2009)

    MathSciNet  MATH  Google Scholar 

  47. Wu, J.-Y.: Upper bounds on the first eigenvalue for a diffusion operator via Bakry–Émery Ricci curvature II. Results Math. 63, 1079–1094 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wu, J.-Y.: \(L^p\)-Liouville theorems on complete smooth metric measure spaces. Bull. Sci. Math. 138, 510–539 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wu, J.-Y., Wu, P.: Heat kernel on smooth metric measure spaces with nonnegative curvature. Math. Ann. 362, 717–742 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Wu, P.: On the potential function of gradient steady Ricci solitons. J. Geom. Anal. 23, 221–228 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhu, S.-H.: The comparison geometry of Ricci curvature. In: Comparison Geometry (Berkeley, CA, 1993–94). Math. Sci. Res. Inst. Publ., vol. 30, pp. 221–262. Cambridge Univ. Press, Cambridge (1997)

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Acknowledgments

The authors thank Professor Laurent Saloff-Coste for his help, and thank Professor Frank Morgan for his suggestions. The work was done when the first named author was visiting the Department of Mathematics at Cornell University, he thanks Professor Xiaodong Cao for his help and the Department of Mathematics for their hospitality. The second named author thanks Professors Xianzhe Dai and Guofang Wei for helpful discussions, constant encouragement and support. The first named author was partially supported by NSFC (11101267, 11271132) and the China Scholarship Council (201208310431). The second named author was partially supported by an AMS-Simons travel grant.

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  1. Department of Mathematics, Shanghai Maritime University, Haigang Avenue 1550, Shanghai, 201306, People’s Republic of China

    Jia-Yong Wu

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Peng Wu

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  1. Jia-Yong Wu
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Wu, JY., Wu, P. Heat kernel on smooth metric measure spaces and applications. Math. Ann. 365, 309–344 (2016). https://doi.org/10.1007/s00208-015-1289-6

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  • DOI: https://doi.org/10.1007/s00208-015-1289-6

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