The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR $3$-manifold
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- by Shu-Cheng Chang, Hung-Lin Chiu and Chin-Tung Wu PDF
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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
- Thierry Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1636569, DOI 10.1007/978-3-662-13006-3
- Bennett Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992), no. 8, 1003–1014. MR 1168117, DOI 10.1002/cpa.3160450805
- Shu-Cheng Chang and Jih-Hsin Cheng, The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3, Ann. Global Anal. Geom. 21 (2002), no. 2, 111–121. MR 1894940, DOI 10.1023/A:1014792304440
- Shu-Cheng Chang, Jih-Hsin Cheng, and Hung-Lin Chiu, A fourth order curvature flow on a CR 3-manifold, Indiana Univ. Math. J. 56 (2007), no. 4, 1793–1826. MR 2354700, DOI 10.1512/iumj.2007.56.3001
- Shu-Cheng Chang and Hung-Lin Chiu, On the estimate of the first eigenvalue of a sublaplacian on a pseudohermitian 3-manifold, Pacific J. Math. 232 (2007), no. 2, 269–282. MR 2366354, DOI 10.2140/pjm.2007.232.269
- S.-C. Chang, H.-L. Chiu and C.-T. Wu, Subgradient Estimate, Eigenvalue Estimates and Li-Yau Inequality in Pseudohermitian $3$-Manifolds, preprint.
- Shu-Cheng Chang, The 2-dimensional Calabi flow, Nagoya Math. J. 181 (2006), 63–73. MR 2210710, DOI 10.1017/S0027763000025678
- Hung-Lin Chiu, Compactness of pseudohermitian structures with integral bounds on curvature, Math. Ann. 334 (2006), no. 1, 111–142. MR 2208951, DOI 10.1007/s00208-005-0709-4
- Hung-Lin Chiu, The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold, Ann. Global Anal. Geom. 30 (2006), no. 1, 81–96. MR 2249615, DOI 10.1007/s10455-006-9033-9
- Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105 (German). MR 1880, DOI 10.1007/BF01450011
- Shu-Cheng Chang and Chin-Tung Wu, On the existence of extremal metrics on complete noncompact 3-manifolds, Indiana Univ. Math. J. 53 (2004), no. 1, 243–268. MR 2048993, DOI 10.1512/iumj.2004.53.2311
- Huai Dong Cao and Shing-Tung Yau, Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields, Math. Z. 211 (1992), no. 3, 485–504. MR 1190224, DOI 10.1007/BF02571441
- Najoua Gamara, The CR Yamabe conjecture—the case $n=1$, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 105–137. MR 1831872, DOI 10.1007/PL00011303
- Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, DOI 10.1090/conm/071/954419
- David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
- —, Extremals for the Sobolev Inequality on the Heisenberg Group and the $CR$ Yamabe Problem, J. Amer. Math. Soc., 1 (1988), 1-13.
- —, Intrinsic $CR$ Normal Coordinates and the $CR$ Yamabe Problem, J. Diff. Geom., 29 (1989), 303-343.
- John M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), no. 1, 411–429. MR 837820, DOI 10.1090/S0002-9947-1986-0837820-2
- John M. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), no. 1, 157–178. MR 926742, DOI 10.2307/2374543
- Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
- 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 367477, DOI 10.1002/cpa.3160270403
- Matthew J. Gursky, Compactness of conformal metrics with integral bounds on curvature, Duke Math. J. 72 (1993), no. 2, 339–367. MR 1248676, DOI 10.1215/S0012-7094-93-07212-2
- Leon Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, 525–571. MR 727703, DOI 10.2307/2006981
- Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
- Hartmut Schwetlick and Michael Struwe, Convergence of the Yamabe flow for “large” energies, J. Reine Angew. Math. 562 (2003), 59–100. MR 2011332, DOI 10.1515/crll.2003.078
- R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71. MR 931204, DOI 10.1007/BF01393992
- Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9, Kinokuniya Book Store Co., Ltd., Tokyo, 1975. MR 0399517
- Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599, DOI 10.4310/jdg/1214434345
- Rugang Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35–50. MR 1258912
Additional Information
- Shu-Cheng Chang
- Affiliation: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, Republic of China
- Email: scchang@math.ntu.edu.tw
- Hung-Lin Chiu
- Affiliation: Department of Mathematics, National Central University, Chung-Li 32054, Taiwan, Republic of China
- Email: hlchiu@math.ncu.edu.tw
- Chin-Tung Wu
- Affiliation: Department of Applied Mathematics, National PingTung University of Education, PingTung 90003, Taiwan, Republic of China
- Email: ctwu@mail.npue.edu.tw
- 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
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 1681-1698
- MSC (2000): Primary 32V20; Secondary 53C44
- DOI: https://doi.org/10.1090/S0002-9947-09-05011-9
- MathSciNet review: 2574873