Convergence of the CR Yamabe flow. (English) Zbl 1497.32013
Summary: We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold \(M\) of real dimension \(2n+1\). We prove convergence of the CR Yamabe flow when \(n=1\) or \(M\) is spherical.
MSC:
| 32V20 | Analysis on CR manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
References:
| [1] | Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9) 55, 269-296 (1976) · Zbl 0336.53033 |
| [2] | Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41, 253-294 (1988) · Zbl 0649.35033 · doi:10.1002/cpa.3160410302 |
| [3] | Bramanti, M., Brandolini, L.: Schauder estimates for parabolic nondivergence operators of Hörmander type. J. Differ. Equ. 234, 177-245 (2007) · Zbl 1113.35033 · doi:10.1016/j.jde.2006.07.015 |
| [4] | Bramanti, M., Brandolini, L.: \[L^p\] Lp estimate for nonvariational hypoelliptic operators with VMO coefficients. Trans. Am. Math. Soc. 352, 781-822 (2000) · Zbl 0935.35037 · doi:10.1090/S0002-9947-99-02318-1 |
| [5] | Brendle, S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differ. Geom. 69, 217-278 (2005) · Zbl 1085.53028 · doi:10.4310/jdg/1121449107 |
| [6] | Brendle, S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541-576 (2007) · Zbl 1130.53044 · doi:10.1007/s00222-007-0074-x |
| [7] | Chang, S.C., Cheng, J.H.: The Harnack estimate for the Yamabe flow on CR manifolds of dimension 3. Ann. Glob. Anal. Geom. 21, 111-121 (2002) · Zbl 1007.53034 · doi:10.1023/A:1014792304440 |
| [8] | Chang, S.C., Chiu, H.L., Wu, C.T.: The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR 3-manifold. Trans. Am. Math. Soc. 362, 1681-1698 (2010) · Zbl 1192.32020 · doi:10.1090/S0002-9947-09-05011-9 |
| [9] | Chanillo, S., Chiu, H.L., Yang, P.: Embeddability for 3-dimensional Cauchy-Riemann manifolds and CR Yamabe invariants. Duke Math. J. 161, 2909-2921 (2012) · Zbl 1271.32040 · doi:10.1215/00127094-1902154 |
| [10] | Cheng, J.H., Chiu, H.L., Yang, P.: Uniformization of spherical CR manifolds. Adv. Math. 255, 182-216 (2014) · Zbl 1288.32051 · doi:10.1016/j.aim.2014.01.002 |
| [11] | Cheng, J.H., Malchiodi, A., Yang, P.: A positive mass theorem in three dimensional Cauchy-Riemann geometry. Adv. Math. 308, 276-347 (2017) · Zbl 1360.32031 · doi:10.1016/j.aim.2016.12.012 |
| [12] | Chow, B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Commun. Pure Appl. Math. 45, 1003-1014 (1992) · Zbl 0785.53027 · doi:10.1002/cpa.3160450805 |
| [13] | Citti, G.: Semilinear Dirichlet problem involving critical exponent for the Kohn Laplacian. Ann. Mat. Pura Appl. (4) 169, 375-392 (1995) · Zbl 0848.35040 · doi:10.1007/BF01759361 |
| [14] | Citti, G., Uguzzoni, F.: Critical semilinear equations on the Heisenberg group: the effect of the topology of the domain. Nonlinear Anal. 46, 399-417 (2001) · Zbl 1199.35094 · doi:10.1016/S0362-546X(00)00138-3 |
| [15] | Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics. 246. Birkhäuser Boston, Boston (2006) · Zbl 1099.32008 |
| [16] | Folland, G.: The tangential Cauchy-Riemann complex on spheres. Trans. Am. Math. Soc. 171, 83-133 (1972) · Zbl 0249.35013 · doi:10.1090/S0002-9947-1972-0309156-X |
| [17] | Folland, G., Stein, E.: Estimates for the \[{\bar{\partial }}_b\]∂¯b complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429-522 (1974) · Zbl 0293.35012 · doi:10.1002/cpa.3160270403 |
| [18] | Gamara, N.: The CR Yamabe conjecture—the case \[n=1\] n=1. J. Eur. Math. Soc. 3, 105-137 (2001) · Zbl 0988.53013 · doi:10.1007/PL00011303 |
| [19] | Gamara, N., Yacoub, R.: CR Yamabe conjecture—the conformally flat case. Pac. J. Math. 201, 121-175 (2001) · Zbl 1054.32020 · doi:10.2140/pjm.2001.201.121 |
| [20] | Habermann, L.: Conformal metrics of constant mass. Calc. Var. Partial Differ. Equ. 12, 259-279 (2001) · Zbl 1037.53014 · doi:10.1007/PL00009914 |
| [21] | Habermann, L., Jost, J.: Green functions and conformal geometry. J. Differ. Geom. 53, 405-433 (1999) · Zbl 1071.53023 · doi:10.4310/jdg/1214425634 |
| [22] | Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. Am. Math. Soc. (Providence, RI) 71, 237-262 (1988) · Zbl 0663.53031 · doi:10.1090/conm/071/954419 |
| [23] | Ho, P.T.: Result related to prescribing pseudo-Hermitian scalar curvature. Int. J. Math. 24, 29 (2013) · Zbl 1267.32034 · doi:10.1142/S0129167X13500201 |
| [24] | Ho, P.T.: The long time existence and convergence of the CR Yamabe flow. Commun. Contemp. Math. 14, 50 (2012) · Zbl 1246.53087 · doi:10.1142/S0219199712500149 |
| [25] | Ho, P.T.: The Webster scalar curvature flow on CR sphere. Part I. Adv. Math. 268, 758-835 (2015) · Zbl 1301.32028 · doi:10.1016/j.aim.2014.10.005 |
| [26] | Jerison, D., Lee, J.M.: Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Am. Math. Soc. 1, 1-13 (1988) · Zbl 0634.32016 · doi:10.1090/S0894-0347-1988-0924699-9 |
| [27] | Jerison, D., Lee, J.M.: Intrinsic CR normal coordinates and the CR Yamabe problem. J. Differ. Geom. 29, 303-343 (1989) · Zbl 0671.32016 · doi:10.4310/jdg/1214442877 |
| [28] | Jerison, D., Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25, 167-197 (1987) · Zbl 0661.32026 · doi:10.4310/jdg/1214440849 |
| [29] | Lee, J.M.: The Fefferman metric and pseudo-Hermitian invariants. Trans. Am. Math. Soc. 296, 411-429 (1986) · Zbl 0595.32026 |
| [30] | Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17, 37-91 (1987) · Zbl 0633.53062 · doi:10.1090/S0273-0979-1987-15514-5 |
| [31] | Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479-495 (1984) · Zbl 0576.53028 · doi:10.4310/jdg/1214439291 |
| [32] | Schwetlick, H., Struwe, M.: Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math. 562, 59-100 (2003) · Zbl 1079.53100 |
| [33] | Simon, L.: Asymptotics for a class of non-linear evolution equations with applications to geometric problems. Ann. Math. (2) 118, 525-571 (1983) · Zbl 0549.35071 · doi:10.2307/2006981 |
| [34] | Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511-517 (1984) · Zbl 0535.35025 · doi:10.1007/BF01174186 |
| [35] | Tanaka, N.: A differential geometric study on strongly pseudo-convex manifolds. Kinokuniya, Tokyo (1975) · Zbl 0331.53025 |
| [36] | Trudinger, N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265-274 (1968) · Zbl 0159.23801 |
| [37] | Wang, W.: Canonical contact forms on spherical CR manifolds. J. Eur. Math. Soc. 5, 245-273 (2003) · Zbl 1048.53018 · doi:10.1007/s10097-003-0050-8 |
| [38] | Webster, S.M.: Pseudohermitian structures on a real hypersurface. J. Differ. Geom. 13, 25-41 (1978) · Zbl 0379.53016 · doi:10.4310/jdg/1214434345 |
| [39] | Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21-37 (1960) · Zbl 0096.37201 |
| [40] | Ye, R.: Global existence and convergence of Yamabe flow. J. Differ. Geom. 39, 35-50 (1994) · Zbl 0846.53027 · doi:10.4310/jdg/1214454674 |
| [41] | Zhang, Y.: The contact Yamabe flow. Ph.D. thesis, University of Hanover (2006) · Zbl 1297.53007 |
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