Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds

Li, Xiaolong; Wang, Kui

doi:10.1090/tran/8434

Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds
HTML articles powered by AMS MathViewer

by Xiaolong Li and Kui Wang PDF

Trans. Amer. Math. Soc. 374 (2021), 8081-8099 Request permission

Abstract:

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed Kähler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact Kähler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are Kähler analogues of well-known results for Riemannian manifolds.

References

Ben Andrews and Julie Clutterbuck, Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable, J. Differential Equations 246 (2009), no. 11, 4268–4283. MR 2517770, DOI 10.1016/j.jde.2009.01.024
Ben Andrews and Julie Clutterbuck, Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J. 58 (2009), no. 1, 351–380. MR 2504416, DOI 10.1512/iumj.2009.58.3756
Ben Andrews and Julie Clutterbuck, Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 24 (2011), no. 3, 899–916. MR 2784332, DOI 10.1090/S0894-0347-2011-00699-1
Ben Andrews and Julie Clutterbuck, Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue, Anal. PDE 6 (2013), no. 5, 1013–1024. MR 3125548, DOI 10.2140/apde.2013.6.1013
Ben Andrews, Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations, Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, Surv. Differ. Geom., vol. 19, Int. Press, Somerville, MA, 2015, pp. 1–47. MR 3381494, DOI 10.4310/SDG.2014.v19.n1.a1
Ben Andrews and Changwei Xiong, Gradient estimates via two-point functions for elliptic equations on manifolds, Adv. Math. 349 (2019), 1151–1197. MR 3946303, DOI 10.1016/j.aim.2019.04.041
Werner Ballmann, Lectures on Kähler manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2006. MR 2243012, DOI 10.4171/025
Rossella Bartolo, Anna Germinario, and Miguel Sánchez, Convexity of domains of Riemannian manifolds, Ann. Global Anal. Geom. 21 (2002), no. 1, 63–83. MR 1889250, DOI 10.1023/A:1014231603588
Dominique Bakry and Zhongmin Qian, Some new results on eigenvectors via dimension, diameter, and Ricci curvature, Adv. Math. 155 (2000), no. 1, 98–153. MR 1789850, DOI 10.1006/aima.2000.1932
Casey Blacker and Shoo Seto, First eigenvalue of the $p$-Laplacian on Kähler manifolds, Proc. Amer. Math. Soc. 147 (2019), no. 5, 2197–2206. MR 3937693, DOI 10.1090/proc/14395
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Xianzhe Dai, Shoo Seto, and Guofang Wei. Fundamental gap estimate for convex domains on sphere – the case $n=2$. Comm. Anal. Geom., to appear, arXiv:1803.01115, 2018.
Vincent Guedj, Boris Kolev, and Nader Yeganefar, A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds, Anal. PDE 6 (2013), no. 5, 1001–1012. MR 3125547, DOI 10.2140/apde.2013.6.1001
Chenxu He, Guofang Wei, and Qi S. Zhang, Fundamental gap of convex domains in the spheres, Amer. J. Math. 142 (2020), no. 4, 1161–1191. MR 4124117, DOI 10.1353/ajm.2020.0033
Atsushi Kasue, On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 31–44. MR 744066
Pawel Kröger, On the spectral gap for compact manifolds, J. Differential Geom. 36 (1992), no. 2, 315–330. MR 1180385
Peter Li, A lower bound for the first eigenvalue of the Laplacian on a compact manifold, Indiana Univ. Math. J. 28 (1979), no. 6, 1013–1019. MR 551166, DOI 10.1512/iumj.1979.28.28075
Peter Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, vol. 134, Cambridge University Press, Cambridge, 2012. MR 2962229, DOI 10.1017/CBO9781139105798
Xiaolong Li, Moduli of continuity for viscosity solutions, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1717–1724. MR 3451247, DOI 10.1090/proc/12850
Xiaolong Li, Modulus of continuity estimates for fully nonlinear parabolic equations, arXiv:2006.16631, 2020.
André Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III, Dunod, Paris, 1958 (French). MR 0124009
Jun Ling and Zhiqin Lu, Bounds of eigenvalues on Riemannian manifolds, Trends in partial differential equations, Adv. Lect. Math. (ALM), vol. 10, Int. Press, Somerville, MA, 2010, pp. 241–264. MR 2648284
Xiaolong Li, Yucheng Tu, and Kui Wang, On a class of quasi-linear operators on smooth metric measure spaces, arXiv:2009.10418, 2020.
Peter Li and Jiaping Wang, Comparison theorem for Kähler manifolds and positivity of spectrum, J. Differential Geom. 69 (2005), no. 1, 43–74. MR 2169582, DOI 10.4310/jdg/1121540339
Xiaolong Li and Kui Wang, Moduli of continuity for viscosity solutions on manifolds, J. Geom. Anal. 27 (2017), no. 1, 557–576. MR 3606561, DOI 10.1007/s12220-016-9690-2
Xiaolong Li and Kui Wang, Sharp lower bound for the first eigenvalue of the weighted $p$-Laplacian I, J. Geom. Anal, to appear, arXiv:1910.02295, 2019.
Xiaolong Li and Kui Wang, Sharp lower bound for the first eigenvalue of the weighted $p$-Laplacian II, Math. Res. Lett, to appear, arXiv:1911.04596, 2019.
Xiaolong Li and Kui Wang, First Robin eigenvalue of the $p$-Laplacian on Riemannian manifolds, Math. Z., to appear, arXiv:2002.06472, 2020.
Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435
Lei Ni, Estimates on the modulus of expansion for vector fields solving nonlinear equations, J. Math. Pures Appl. (9) 99 (2013), no. 1, 1–16 (English, with English and French summaries). MR 3003280, DOI 10.1016/j.matpur.2012.05.009
Aaron Naber and Daniele Valtorta, Sharp estimates on the first eigenvalue of the $p$-Laplacian with negative Ricci lower bound, Math. Z. 277 (2014), no. 3-4, 867–891. MR 3229969, DOI 10.1007/s00209-014-1282-x
Lei Ni, Qingsong Wang, and Fangyang Zheng. Manifolds with positive orthogonal ricci curvature. arXiv:1806.10233v2, 2018.
Lei Ni and Fangyang Zheng, Comparison and vanishing theorems for Kähler manifolds, Calc. Var. Partial Differential Equations 57 (2018), no. 6, Paper No. 151, 31. MR 3858834, DOI 10.1007/s00526-018-1431-x
Lei Ni and Fangyang Zheng, On orthogonal Ricci curvature, Advances in complex geometry, Contemp. Math., vol. 735, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 203–215. MR 3996490, DOI 10.1090/conm/735/14827
Robert C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), no. 3, 459–472. MR 474149, DOI 10.1512/iumj.1977.26.26036
Shoo Seto, Lili Wang, and Guofang Wei, Sharp fundamental gap estimate on convex domains of sphere, J. Differential Geom. 112 (2019), no. 2, 347–389. MR 3960269, DOI 10.4310/jdg/1559786428
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
Yôtarô Tsukamoto, On Kählerian manifolds with positive holomorphic sectional curvature, Proc. Japan Acad. 33 (1957), 333–335. MR 90845
Luen-Fai Tam and Chengjie Yu, Some comparison theorems for Kähler manifolds, Manuscripta Math. 137 (2012), no. 3-4, 483–495. MR 2875289, DOI 10.1007/s00229-011-0477-2
Hajime Urakawa, Stability of harmonic maps and eigenvalues of the Laplacian, Trans. Amer. Math. Soc. 301 (1987), no. 2, 557–589. MR 882704, DOI 10.1090/S0002-9947-1987-0882704-8
Daniele Valtorta, Sharp estimate on the first eigenvalue of the $p$-Laplacian, Nonlinear Anal. 75 (2012), no. 13, 4974–4994. MR 2927560, DOI 10.1016/j.na.2012.04.012
H. Wu, An elementary method in the study of nonnegative curvature, Acta Math. 142 (1979), no. 1-2, 57–78. MR 512212, DOI 10.1007/BF02395057
Yuntao Zhang and Kui Wang, An alternative proof of lower bounds for the first eigenvalue on manifolds, Math. Nachr. 290 (2017), no. 16, 2708–2713. MR 3722506, DOI 10.1002/mana.201600388
Jia Qing Zhong and Hong Cang Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984), no. 12, 1265–1273. MR 794292