Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR $ 3$-manifold

Authors: Shu-Cheng Chang, Hung-Lin Chiu and Chin-Tung Wu
Journal: Trans. Amer. Math. Soc. 362 (2010), 1681-1698
MSC (2000): Primary 32V20; Secondary 53C44
Published electronically: November 17, 2009
MathSciNet review: 2574873
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We deform the contact form by the (normalized) CR Yamabe flow on a closed spherical CR $ 3$-manifold. We show that if a contact form evolves with positive Tanaka-Webster curvature and vanishing torsion from initial data, then we obtain a new Li-Yau-Hamilton inequality for the CR Yamabe flow. By combining this parabolic subgradient estimate with a compactness theorem of a sequence of contact forms, it follows that the CR Yamabe flow exists for all time and converges smoothly to, up to the CR automorphism, a unique limit contact form of positive constant Webster scalar curvature on a closed CR $ 3$-manifold, which is CR equivalent to the standard CR $ 3$-sphere with positive Tanaka-Webster curvature and vanishing torsion.

References [Enhancements On Off] (What's this?)

  • [A] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569 (99i:58001)
  • [C] B. Chow, The Yamabe Flow on Locally Conformally Flat Manifolds with Positive Ricci Curvature, Comm. Pure Appl. 45(1992), 1003-1014. MR 1168117 (93d:53045)
  • [CC] S.-C. Chang and J.-H. Cheng, The Harnack Estimate for the Yamabe Flow on CR Manifolds of Dimension 3, AGAG, 21 (2002), 111-121. MR 1894940 (2002m:53055)
  • [CCC] S.-C. Chang, J.-H. Cheng and H.-L. Chiu, The Fourth-order Q-curvature Flow on a CR 3-Manifold, Indiana Univ. Math. J., 56, No. 4 (2007), 1793-1826. MR 2354700 (2009a:53111)
  • [CCh] S.-C. Chang and H.-L. Chiu, On the Estimate of First Eigenvalue of a Sublaplacian on a Pseudo-Hermitian $ 3$-Manifold, Pacific Journal of Mathematics, 232, No. 2 (2007), 269-282. MR 2366354 (2008m:58065)
  • [CCW] S.-C. Chang, H.-L. Chiu and C.-T. Wu, Subgradient Estimate, Eigenvalue Estimates and Li-Yau Inequality in Pseudohermitian $ 3$ -Manifolds, preprint.
  • [Ch] S.-C. Chang, The 2-Dimensional Calabi Flow, Nagoya Math. J., 181 (2006), 63-73. MR 2210710 (2006j:53095)
  • [Chi1] H.-L. Chiu, Compactness of Pseudohermitian Structures with Integral Bounds on Curvature, Math. Ann., 334 (2006), 111-142. MR 2208951 (2007d:32028)
  • [Chi2] H.-L. Chiu, The Sharp Lower Bound for the First Positive Eigenvalue of the Sublaplacian on a Pseudohermitian 3-Manifold, Annals of Global Analysis and Geometry, 30 (2006), 81-96. MR 2249615 (2007j:58034)
  • [Cho] W.-L. Chow, Uber System Von Lineaaren Partiellen Differentialgleichungen erster Orduung, Math. Ann., 117 (1939), 98-105. MR 0001880 (1:313d)
  • [CW] S.-C. Chang and C.-T. Wu, On the Existence of Extremal Metrics on Complete Noncompact $ 3$-Manifolds, Indiana Univ. Math. J., 53 (2004), 243-268. MR 2048993 (2005a:53057)
  • [CY] H.-D. Cao and S.-T. Yau, Gradient Estimates, Harnack Inequalities and Estimates for Heat Kernels of the Sum of Squares of Vector Fields, Math. Z., 211 (1992), 485-504. MR 1190224 (94h:58155)
  • [GY] N. Gamara and Y. Yacoub, The CR Yamabe conjecture-the case $ n=1$ , J. Eur. Math. Soc. 3 (2001), 105-137. MR 1831872 (2003d:32040a)
  • [H] R. S. Hamilton, The Ricci Flow on Surfaces, Mathematics and General Relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1998. MR 954419 (89i:53029)
  • [JL1] D. Jerison and J. M. Lee, The Yamabe Problem on $ CR$ manifolds, J. Diff. Geom., 25 (1987), 167-197. MR 880182 (88i:58162)
  • [JL2] -, Extremals for the Sobolev Inequality on the Heisenberg Group and the $ CR$ Yamabe Problem, J. Amer. Math. Soc., 1 (1988), 1-13.
  • [JL3] -, Intrinsic $ CR$ Normal Coordinates and the $ CR $ Yamabe Problem, J. Diff. Geom., 29 (1989), 303-343.
  • [L1] J. M. Lee, The Fefferman Metric and Pseudohermitian Invariants, Trans. Amer. Math. Soc., 296 (1986), 411-429. MR 837820 (87j:32063)
  • [L2] -, Pseudo-Einstein Structures on $ CR$ Manifolds, Am. J. Math., 110 (1988), 157-178. MR 0926742 (89f:32034)
  • [LY] P. Li and S.-T. Yau, On the Parabolic Kernel of the Schr ődinger Operator, Acta Math., 156 (1985), 153-201. MR 834612 (87f:58156)
  • [Fo] G. B. Folland, Subelliptic Estimates and Function Spaces on Nilpotent Lie Groups, Ark. Mat. 13 (1975), 161-207. MR 0494315 (58:13215)
  • [FS] G. B. Folland and E. M. Stein, Estimates for the $ \bar{ \partial}_{b}$ Complex and Analysis on the Heisenberg Group, Comm. Pure Appl. Math., 27 (1974), 429-522. MR 0367477 (51:3719)
  • [Gu] M.J. Gursky, Compactness of Conformal Metrics with Integral Bounds on Curvature, Duke Math. J. Vol. 72, No. 2 (1993), 339-367. MR 1248676 (94m:53054)
  • [S] L. Simon, Asymptotics for a Class of Nonlinear Evolution Equations, with Applications to Geometric Problems, Ann. of Math., 118 (1983), 525-571. MR 727703 (85b:58121)
  • [Sch] R. Schoen, Conformal Deformation of a Riemannian Metric to a Constant Scalar Curvature, J. Diff. Geom., 20 (1984), 479-495. MR 788292 (86i:58137)
  • [SS] H. Schwetlick and M. Struwe, Convergence of the Yamabe Flow for Large Energies, J. Reine Angew. Math., 562 (2003), 59-100. MR 2011332 (2004h:53097)
  • [SY] R. Schoen and S.-T. Yau, Conformally Flat, Kleinian Groups and Scalar Curvature, Invent. Math. 92 (1988), 47-71. MR 931204 (89c:58139)
  • [T] N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifolds, 1975, Kinokuniya Co. Ltd., Tokyo. MR 0399517 (53:3361)
  • [Tru] N. Trudinger, Remarks Concerning the Conformal Deformation of Riemannian Structures on Compact Manifolds, Ann. Scuola Norm. Sup. Pisa CI. Sci. (4), 22 (1968), 265-274. MR 0240748 (39:2093)
  • [W] S. M. Webster, Pseudohermitian Structures on a Real Hypersurface, J. Diff. Geom., 13 (1978), 25-41. MR 520599 (80e:32015)
  • [Y] R. Ye, Global Existence and Convergence of Yamabe Flow, J. Diff. Geom., 39 (1994), 35-50. MR 1258912 (95d:53044)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32V20, 53C44

Retrieve articles in all journals with MSC (2000): 32V20, 53C44

Additional Information

Shu-Cheng Chang
Affiliation: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, Republic of China

Hung-Lin Chiu
Affiliation: Department of Mathematics, National Central University, Chung-Li 32054, Taiwan, Republic of China

Chin-Tung Wu
Affiliation: Department of Applied Mathematics, National PingTung University of Education, PingTung 90003, Taiwan, Republic of China

Keywords: Li-Yau-Hamilton inequality, CR Bochner formula, Tanaka-Webster curvature, pseudoharmonic manifold, CR pluriharmonic operator, CR Paneitz operator, sub-Laplacian, subgradient estimate, CR Yamabe flow, positive mass theorem.
Received by editor(s): January 23, 2007
Published electronically: November 17, 2009
Additional Notes: This research was supported in part by the NSC of Taiwan
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society