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Improved Beckner-Sobolev inequalities on Kähler manifolds. (English) Zbl 1467.53073

This paper concerns the best constant in the Sobolev inequality for compact Kähler manifolds with the Ricci curvature bounded below by a positive constant and the normalised volume. The main result gives an explicit constant in terms of the dimension of the manifold and the lower bound of the Ricci curvature. The authors then apply this theorem to improve the diameter bound for Kähler manifolds given by the Bonnet-Myers theorem.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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[1] Bakry, D.; Ledoux, M., Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator, Duke Math. J., 85, 1, 253-270 (1996) · Zbl 0870.60071 · doi:10.1215/S0012-7094-96-08511-7
[2] Bakry, D.; Gentil, I.; Ledoux, M., Analysis and Geometry of Markov Diffusion Operators (2013), Cham: Springer, Cham · Zbl 1376.60002
[3] Baudoin, F.: Geometric Inequalities on Riemannian and sub-Riemannian manifolds by heat semigroups techniques. arXiv:1801.05702
[4] Bidaut-Veron, M-F; Veron, L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106, 489-539 (1991) · Zbl 0755.35036 · doi:10.1007/BF01243922
[5] Cheng, SY, Eigenvalue comparison theorems and its geometric application, Math. Z., 143, 289-297 (1975) · Zbl 0329.53035 · doi:10.1007/BF01214381
[6] Dolbeault, J.; Esteban, MJ; Kowalczyk, M.; Loss, M., Sharp interpolation inequalities on the sphere: new methods and consequences, Chin. Ann. Math. B, 34, 1, 99-112 (2013) · Zbl 1263.26029 · doi:10.1007/s11401-012-0756-6
[7] Gentil, I., Zugmeyer, A.: A family of Beckner inequalities under various curvature-dimension conditions. arXiv:1903.00214
[8] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34, 525-598 (1981) · Zbl 0465.35003 · doi:10.1002/cpa.3160340406
[9] Hebey, E., Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities (1999), New York: Courant Institute of Mathematical Sciences, New York · Zbl 0981.58006
[10] Hebey, E.; Vaugon, M., Meilleures constantes dans le theoreme d’inclusion de Sobolev, Ann. l’Inst. Henri Poincaré, 13, 1, 57-93 (1996) · Zbl 0849.53035 · doi:10.1016/S0294-1449(16)30097-X
[11] Ledoux, M., The geometry of Markov diffusion generators. Probability theory, Ann. Fac. Sci. Toulouse Math. (6), 9, 2, 305-366 (2000) · Zbl 0980.60097 · doi:10.5802/afst.962
[12] Li, P.; Wang, J., Comparison theorem for Kähler manifolds and positivity of spectrum, J. Differ. Geom., 69, 43-74 (2005) · Zbl 1087.53067 · doi:10.4310/jdg/1121540339
[13] Liu, G., Kähler manifolds with Ricci curvature lower bound, Asian J. Math., 18, 1, 69-99 (2014) · Zbl 1306.53023 · doi:10.4310/AJM.2014.v18.n1.a4
[14] Liu, G.; Yuan, Y., Diameter rigidity for Kähler manifolds with positive bisectional curvature, Math. Z., 290, 1055-1061 (2018) · Zbl 1401.53061 · doi:10.1007/s00209-018-2052-y
[15] Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differ. Geom. 6, 247-258 (1971/1972) · Zbl 0236.53042
[16] Saloff-Coste, L., Precise estimates on the rate at which certain diffusions tend to equilibrium, Math. Z., 217, 641-677 (1994) · Zbl 0815.60074 · doi:10.1007/BF02571965
[17] Tam, LF; Yu, C., Some comparison theorems for Kähler manifolds, Manuscr. Math., 137, 3-4, 483-495 (2012) · Zbl 1243.53114 · doi:10.1007/s00229-011-0477-2
[18] Tsukamoto, Y., On Kählerian manifolds with positive holomorphic sectional curvature, Proc. Jpn Acad., 33, 333-335 (1957) · Zbl 0078.14205 · doi:10.3792/pja/1195525029
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