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A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth

  • Der-Chen Chang (a1) (a2), Shu-Cheng Chang (a3), Yingbo Han (a4) and Jingzhi Tie (a5)

Abstract

In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$ -manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

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Author D.-C. C. is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. Author S.-C. C. is partially supported by the MOST of Taiwan. (Corresponding author) Y. H. is partially supported by an NSFC grant 11201400, Nanhu Scholars Program for Young Scholars of Xinyang Normal University and the Universities Young Teachers Program of Henan Province (2016GGJS-096).

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[1] Bakry, D. and Emery, M., Diffusions hypercontractives. In: Séminaire de probabilité, XIX. Lecture Notes in Math., 1123. Springer, Berlin, 1985, pp. 177–206. https://doi.org/10.1007/BFb0075847.
[2] Baudoin, F., Bonnefont, M., and Garofalo, N., A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality . Math. Ann. 358(2014), no. 3–4, 833860. https://doi.org/10.1007/s00208-013-0961-y.
[3] Baudoin, F., Bonnefont, M., Garofalo, N., and Munive, I., Volume and distance comparison theorems for sub-Riemannian manifolds . J. Funct. Anal. 267(2014), no. 7, 20052027. https://doi.org/10.1016/j.jfa.2014.07.030.
[4] Beals, R. and Greiner, P. C., Calculus on Heisenberg manifolds. Annals of Mathematics Studies, 119. Princeton University Press, Princeton, NJ, 1988. https://doi.org/10.1515/9781400882397.
[5] Beals, R., Gaveau, B., and Greiner, P. C., Hamilton-Jacobi theory and the heat kernel on Heisenberg groups . J. Math Pures Appl. 79(2000), 633689. https://doi.org/10.1016/S0021-7824(00)00169-0.
[6] Beardon, A. F., Sums of powers of integers . Amer. Math. Monthly 103(1996), no. 3, 201213. https://doi.org/10.2307/2975368.
[7] Berenstein, C., Chang, D.-C., and Tie, J., Laguerre calculus and its application on the Heisenberg group. AMS/IP Studies in Advanced Mathematics, 22. American Mathematical Society, Providence, RI and International Press, CambridgeSomerville, MA, 2001.
[8] Calin, O., Chang, D.-C., Furutani, K., and Iwasaki, C., Heat kernels for elliptic and sub-elliptic operators: methods and techniques. Birkhäuser-Springer, New York, 2011. https://doi.org/10.1007/978-0-8176-4995-1.
[9] Calin, O., Chang, D.-C., and Tie, J., Fundamental solutions for Hermite and subelliptic operators . J. Anal. Math. 100(2006), 223248. https://doi.org/10.1007/BF02916762.
[10] Cao, H.-D. and Yau, S.-T., Gradient estimate, Harnack inequalities and estimates for Heat kernel of the sum of squares of vector fields . Math. Z. 221(1992), 485504. https://doi.org/10.1007/BF02571441.
[11] Chang, D.-C., Chang, S.-C., and Lin, C., On Li-Yau gradient estimate for sum of squares of vector fields up to higher step. To appear in: Comm. Anal. Geom. arxiv:1602.01531 https://doi.org/10.1007/BF02571441.
[12] Chang, S.-C., Kuo, T.-J., and Lai, S.-H., Li-Yau gradient estimate and entropy formulae for the CR heat equation in a closed pseudohermitian 3-manifold . J. Differential Geom. 89(2011), 185216. https://doi.org/10.4310/jdg/1324477409.
[13] Chang, S.-C., Kuo, T.-J., Lin, C., and Tie, J., CR sub-Laplacian comparison and Liouville-type theorem in a complete noncompact Sasakian manifold. arxiv:1506.03270.
[14] Chang, S.-C., Tie, J., and Wu, C.-T., Subgradient estimate and Liouville-type theorem for the CR heat equation on Heisenberg groups . Asian J. Math. 14(2010), no. 1, 4172. https://doi.org/10.4310/AJM.2010.v14.n1.a4.
[15] Cheeger, J., Colding, T. H., and Minicozzi, W. P., Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature . Geom. Funct. Anal. 5(1995), no. 6, 948954. https://doi.org/10.1007/BF01902216.
[16] Cheng, S.-Y. and Yau, S.-T., Differential equations on Riemannian manifolds and their geometric applications . Comm. Pure Appl. Math. 28(1975), 333354. https://doi.org/10.1002/cpa.3160280303.
[17] Chow, W.-L., Uber system von lineaaren partiellen Differentialgleichungen erster Orduung . Math. Ann. 117(1939), 98105. https://doi.org/10.1007/BF01450011.
[18] Cohn, W. S. and Lu, G., Best constants for Moser-Trudinger inequalities on the Heisenberg group . Indiana Univ. Math. J. 50(2001), 15671591. https://doi.org/10.1512/iumj.2001.50.2138.
[19] Colding, T. H. and Minicozzi, W. P., Harmonic functions on manifolds . Ann. of Math. 146(1997), no. 3, 725747. https://doi.org/10.2307/2952459.
[20] Dragomir, S. and Tomassini, G., Differential geometry and analysis on CR manifolds. Progress in Mathematics, 246. Birkhäuser Boston, Boston, MA, 2006.
[21] Dunkl, C. F., An addition theorem for Heisenberg harmonics . In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth International, Belmont, CA, 1983, pp. 690707.
[22] Folland, G. B. and Stein, E. M., Estimates for the ̄ b complex and analysis on the Heisenberg group . Comm. Pure Appl. Math. 27(1974), 429522. https://doi.org/10.1002/cpa.3160270403.
[23] Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents . Acta Math. 139(1977), 95153. https://doi.org/10.1007/BF02392235.
[24] Greenleaf, A., The first eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold . Comm. Partial Differential Equations 10(1985), no. 3, 191217. https://doi.org/10.1080/03605308508820376.
[25] Greiner, P. C., Spherical harmonics on the Heisenberg group . Canad. Math. Bull. 23(1980), 383396. https://doi.org/10.4153/CMB-1980-057-9.
[26] Gromov, M., Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics, 152. Birkhäuser Boston, Boston, MA, 1999.
[27] Hulanicki, A., The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group . Studia Math. 56(1976), 165173. https://doi.org/10.4064/sm-56-2-165-173.
[28] Kasue, A., Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature II . In: Recent topics in differential and analytic geometry. Adv. Stud. Pure Math., 18. Academic Press, Boston, MA, 1990, pp. 283301.
[29] Korányi, A. and Stanton, N., Liouville-type theorems for some complex hypoelliptic operators . J. Funct. Anal. 60(1985), 370377. https://doi.org/10.1016/0022-1236(85)90045-X.
[30] Lee, J. M., Pseudo-Einstein structure on CR manifolds . Amer. J. Math. 110(1988), 157178. https://doi.org/10.2307/2374543.
[31] Li, P., Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press, Cambridge, 2012. https://doi.org/10.1017/CBO9781139105798.
[32] Li, P., Harmonic functions on Kähler manifolds with nonnegative Ricci curvature . Math. Res. Lett. 2(1995), 7994. https://doi.org/10.4310/MRL.1995.v2.n1.a8.
[33] Li, P., Harmonic sections of polynomial growth . Math. Res. Lett. 4(1997), 3544. https://doi.org/10.4310/MRL.1997.v4.n1.a4.
[34] Li, P. and Tam, L.-F., Linear growth harmonic functions on a complete manifold . J. Differential Geom. 29(1989), 421425. https://doi.org/10.4310/jdg/1214442883.
[35] Li, P. and Tam, L.-F., Complete surfaces with finite total curvature . J. Differential Geom. 33(1991), 139168. https://doi.org/10.4310/jdg/1214446033.
[36] Li, P. and Yau, S.-T., On the parabolic kernel of the Schrödinger operator . Acta Math. 156(1985), 153201. https://doi.org/10.1007/BF02399203.
[37] Liu, G., On the volume growth of Kähler manifolds with nonnegative bisectional curvature . J. Differential Geom. 102(2016), no. 3, 485500. https://doi.org/10.4310/jdg/1456754016.
[38] Nagel, A., Analysis and geometry on Carnot-Carathéodory spaces. http://www.math.wisc.edu/∼nagel/2005Book.pdf.
[39] Saloff-Coste, L., Uniformly elliptic operators on Riemannian manifolds . J. Differential Geom. 36(1992), 417450. https://doi.org/10.4310/jdg/1214448748.
[40] Saloff-Coste, L., Aspect of Sobolev-type inequalities. London Mathematical Society Lecture Notes Series, 289. Cambridge University Press, Cambridge, 2002.
[41] Siu, Y.-T. and Yau, S.-T., Complete Kähler manifolds with non-positive curvature of faster than quadratic decay . Ann. of Math. 105(1977), 225264. https://doi.org/10.2307/1970998.
[42] Villani, C., Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin, 2009. https://doi.org/10.1007/978-3-540-71050-9.
[43] Xu, G., Three circles theorems for harmonic functions . Math. Ann. 366(2016), 12811317. https://doi.org/10.1007/s00208-016-1366-5.
[44] Yau, S.-T., Harmonic functions on complete Riemannian manifolds . Comm. Pure Appl. Math. 28(1975), 201228. https://doi.org/10.1002/cpa.3160280203.
[45] Yau, S.-T., Open problems in geometry. In: Lectures on differential geometry, International Press, 1994, pp. 365–404.
[46]S.-T. Yau, ed. Seminar on differential geometry. Annals of Mathematics Studies, 102. Princeton University Press, Princeton, NJ, 1982.
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