Abstract
In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and the decay rates of L 1 and L 2 energy for drifting heat equation on closed Riemannian manifolds with weighted measure.
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Aubin, T.: Some Nonlinear Problems in Riemannian Geometry, Springer Monogr. Math., Springer-Verlag, Berlin, 1998
Bakry, D., Emery, M.: Diffusion hypercontractivitives, in Seminaire de Probabilites XIX, 1983/1984, 177–206, Lect. Notes in Math. 1123, Springer, Berlin, 1985
Bakry, D., Qian, Z. M.: Volume comparison theorems without Jacobi fields. Current trends in potential theory, 115–122, Theta Ser. Adv. Math., 4, Theta, Bucharest, 2005
Chow, B., Hamilton, R.: Constrained and linear Harnack inequalities for parabolic equations. Inventiones Mathematicae, 129, 213–238 (1997)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci flow, Lectures in Contemporary Mathematics 3, Science Press and American Mathematical Society, Beijing, 2006
Dai, X. Z., Ma, L.: Mass under Ricci flow. Commun. Math. Phys., 274, 65–80 (2007)
Gonzalez, B., Negrin, E.: Gradient estimates for positive solutions of the Laplacian with drift. Proc. Amer. Math. Soc., 127, 619–625 (1999)
Evans, L.: Partial Differential Equations, Graduate studies in Math., AMS, Providence, Rhode Island, 1986
Hamilton, R.: The formation of Singularities in the Ricci flow. Surveys in Diff. Geom., 2, 7–136 (1995)
Li, P., Yau, S. T.: On the parabolic kernel of the Schrödinger operator. Acta Math., 156, 153–201 (1986)
Li, J. Y.: Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J. Funct. Anal., 100, 233–256 (1991)
Li, X. D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pure. Appl., 84, 1295–1361 (2005)
Limoncu, M.: The Bochner technique and modification of the Ricci tensor. Ann. Glob. Anal. Geom., 36, 285–291 (2009)
Ling, J.: A lower bound of the first eigenvalue of a closed manifold with positive Ricci curvature. Ann. Global Anal. Geom., 31(4), 385–408 (2007)
Ma, L.: Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds. J. Funct. Anal., 241, 374–382 (2006)
Ma, L., Du, S. H.: Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. C. R. Acad. Sci. Paris, Ser. I, 348, 1203–1206 (2010)
Ma, L., Liu, B. Y.: Convex eigenfunction of a drifting Laplacian operator and the fundamental gap. Pacific J. Math., 240, 343–361 (2009)
Ma, L., Liu, B. Y.: Convexity of the first eigenfunction of the drifting Laplacian operator and its applications. New York J. Math., 14, 393–401 (2008)
Qian, Z. M.: An estimate for the vorticity of the Navier-Stokes equation. Comptes Rendus Mathematique, 347(1-2), 89–92 (2009)
Schoen, R., Yau, S. T.: Lectures on Differential Geometry, International Press, 1994
Struwe, M.: Variatonal Methods, 3rd edition, Springer, 2000
Xia, C. Y.: Universal inequality for eigenvalues of the vibration problem for a clamped plate on Riemannian manifolds. Quarterly J. Math., 62(1), 235–258 (2011)
Yang, Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifold. Proc. Amer. Math. Soc., 136, 4095–4102 (2008)
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Supported by National Natural Science Foundation of China (Grant No. 11271111)
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Ma, L. Eigenvalue estimates and L 1 energy on closed manifolds. Acta. Math. Sin.-English Ser. 30, 1729–1734 (2014). https://doi.org/10.1007/s10114-014-1726-6
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DOI: https://doi.org/10.1007/s10114-014-1726-6