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Eigenvalue Estimates on Quaternion-Kähler Manifolds

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Abstract

We prove lower bound estimates for the first nonzero eigenvalue of the Laplacian on a compact quaternion-Kähler manifold. For the closed or Neumann case, the lower bounds depend on dimension, diameter, and lower bound of scalar curvature, and they are derived as the large time implication of the modulus of continuity estimates for solutions of the heat equation. For the Dirichlet case, we establish lower bounds that depend on dimension, inradius, lower bound of scalar curvature, and lower bound of the second fundamental form of the boundary, via a Laplace comparison theorem for the distance to the boundary function.

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References

  1. Andrews, B., Clutterbuck, J.: Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable. J. Differ. Equ. 246(11), 4268–4283 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, B., Clutterbuck, J.: Time-interior gradient estimates for quasilinear parabolic equations. Indiana Univ. Math. J. 58(1), 351–380 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrews, B., Clutterbuck, J.: Proof of the fundamental gap conjecture. J. Am. Math. Soc. 24(3), 899–916 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andrews, B., Clutterbuck, J.: Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue. Anal. PDE 6(5), 1013–1024 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alekseevsky, D.V., Marchiafava, S.: Transformations of a quaternionic Kähler manifold. C. R. Acad. Sci. Paris Sér. I Math. 320(6), 703–708 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Andrews, B.: Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations. In: Surveys in Differential Geometry 2014. Regularity and Evolution of Nonlinear Equations, vol. 19, pp. 1–47. Int. Press, Somerville (2015)

  7. Andrews, B., Ni, L.: Eigenvalue comparison on Bakry–Emery manifolds. Commun. Partial Differ. Equ. 37(11), 2081–2092 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Andrews, B., Xiong, C.: Gradient estimates via two-point functions for elliptic equations on manifolds. Adv. Math. 349, 1151–1197 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bakry, D., Qian, Z.: Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Adv. Math. 155(1), 98–153 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ballmann, W.: Lectures on Kähler Manifolds. ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich, (2006)

  11. Baudoin, F., Yang, G.: Brownian motions and heat kernel lower bounds on Kähler and quaternion Kähler manifolds. Int. Math. Res. Not. IMRN 6, 4659–4681 (2022)

    Article  MATH  Google Scholar 

  12. Berger, M.: Remarques sur les groupes d’holonomie des variétés riemanniennes. C. R. Acad. Sci. Paris Sér. A-B 262, A1316–A1318 (1966)

    MATH  Google Scholar 

  13. Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008). Reprint of the 1987 edition

  14. Blacker, C., Seto, S.: First eigenvalue of the \(p\)-Laplacian on Kähler manifolds. Proc. Am. Math. Soc. 147(5), 2197–2206 (2019)

    Article  MATH  Google Scholar 

  15. Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115. Academic Press, Orlando (1984). Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk

  16. Chen, M.F., Wang, F.Y.: Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A 37(1), 1–14 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dai, X., Seto, S., Wei, G.: Fundamental gap estimate for convex domains on sphere-the case \(n = 2\). Commun. Anal. Geom. 29(5), 1095–1125 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Futaki, A., Li, H., Li, X.-D.: On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons. Ann. Glob. Anal. Geom. 44(2), 105–114 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guedj, V., Kolev, B., Yeganefar, N.: A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds. Anal. PDE 6(5), 1001–1012 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hang, F., Wang, X.: A remark on Zhong–Yang’s eigenvalue estimate. Int. Math. Res. Not. IMRN, 2007(18), Art. ID rnm064, 9 (2007)

  22. He, C., Wei, G., Zhang, Q.S.: Fundamental gap of convex domains in the spheres. Am. J. Math. 142(4), 1161–1192 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ishihara, S.: Quaternion Kählerian manifolds. J. Differ. Geom. 9, 483–500 (1974)

    Article  MATH  Google Scholar 

  24. Kasue, A.: On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. Ann. Sci. École Norm. Sup. (4) 17(1), 31–44 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kong, S., Li, P., Zhou, D.: Spectrum of the Laplacian on quaternionic Kähler manifolds. J. Differ. Geom. 78(2), 295–332 (2008)

    Article  MATH  Google Scholar 

  26. Kröger, P.: On the spectral gap for compact manifolds. J. Differ. Geom. 36(2), 315–330 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, P.: A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana Univ. Math. J. 28(6), 1013–1019 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, X.: Moduli of continuity for viscosity solutions. Proc. Am. Math. Soc. 144(4), 1717–1724 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, X.: Modulus of continuity estimates for fully nonlinear parabolic equations. Calc. Var. Partial Differ. Equ. 60(5), Paper No. 182, 23 (2021)

  30. Li, P., Wang, J.: Comparison theorem for Kähler manifolds and positivity of spectrum. J. Differ. Geom. 69(1), 43–74 (2005)

    Article  MATH  Google Scholar 

  31. Li, X., Wang, K.: Moduli of continuity for viscosity solutions on manifolds. J. Geom. Anal. 27(1), 557–576 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, X., Wang, K.: First Robin eigenvalue of the \(p\)-Laplacian on Riemannian manifolds. Math. Z. 298(3–4), 1033–1047 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, X., Wang, K.: Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds. Trans. Am. Math. Soc. 374(11), 8081–8099 (2021)

    MATH  Google Scholar 

  34. Li, X., Wang, K.: Sharp lower bound for the first eigenvalue of the weighted \(p\)-Laplacian I. J. Geom. Anal. 31(8), 8686–8708 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, X., Wang, K.: Sharp lower bound for the first eigenvalue of the weighted \(p\)-Laplacian II. Math. Res. Lett. 28(5), 1459–1479 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, X., Tu, Y., Wang, K.: On a class of quasi-linear operators on smooth metric measure spaces (2020). arXiv:2009.10418

  37. Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace Operator (Proceedings of Symposia in Pure Mathematics, University of Hawaii, Honolulu, Hawaii, 1979), vol. XXXVI, pp. 205–239. American Mathematical Society, Providence (1980)

  38. Lichnerowicz, A.: Géométrie des groupes de transformations. Travaux et Recherches Mathematiques, III. Dunod, Paris (1958)

  39. Ling, J., Lu, Z.: Bounds of eigenvalues on Riemannian manifolds. In: Trends in Partial Differential Equations. Advanced Lectures in Mathematics (ALM), vol. 10, pp. 241–264. Int. Press, Somerville (2010)

  40. Liu, G.: Kähler manifolds with Ricci curvature lower bound. Asian J. Math. 18(1), 69–99 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ni, L.: Estimates on the modulus of expansion for vector fields solving nonlinear equations. J. Math. Pures Appl. (9) 99(1), 1–16 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ni, L., Zheng, F.: Comparison and vanishing theorems for Kähler manifolds. Calc. Var. Partial Differ. Equ. 57(6), Art. 151, 31 (2018)

  43. Ni, L., Zheng, F.: On orthogonal Ricci curvature. In: Advances in Complex Geometry. Contemporary Mathematics, vol. 735, pp. 203–215. American Mathematical Society, Providence (2019)

  44. Ni, L., Wang, Q., Zheng, F.: Manifolds with positive orthogonal Ricci curvature. Am. J. Math. 143(3), 833–857 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  45. Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rutkowski, B., Seto, S.: Explicit lower bound of the first eigenvalue of the Laplacian on Kähler manifolds. Involve, to appear (2022). arXiv:2207.10966

  47. Seto, S., Wang, L., Wei, G.: Sharp fundamental gap estimate on convex domains of sphere. J. Differ. Geom. 112(2), 347–389 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge (1994)

  49. Shi, Y.M., Zhang, H.C.: Lower bounds for the first eigenvalue on compact manifolds. Chin. Ann. Math. Ser. A 28(6), 863–866 (2007)

    MathSciNet  MATH  Google Scholar 

  50. Tam, L.-F., Chengjie, Y.: Some comparison theorems for Kähler manifolds. Manuscr. Math. 137(3–4), 483–495 (2012)

    Article  MATH  Google Scholar 

  51. Urakawa, H.: Stability of harmonic maps and eigenvalues of the Laplacian. Trans. Am. Math. Soc. 301(2), 557–589 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wu, H.: An elementary method in the study of nonnegative curvature. Acta Math. 142(1–2), 57–78 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhang, Y., Wang, K.: An alternative proof of lower bounds for the first eigenvalue on manifolds. Math. Nachr. 290(16), 2708–2713 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zhong, J.Q., Yang, H.C.: On the estimate of the first eigenvalue of a compact Riemannian manifold. Sci. Sin. Ser. A 27(12), 1265–1273 (1984)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, KS, 67260, USA

    Xiaolong Li

  2. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Kui Wang

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  1. Xiaolong Li
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  2. Kui Wang
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The author X. Li research is partially supported by Simons Collaboration Grant #962228 and a start-up grant at Wichita State University. The research of the second author is supported by NSFC No. 11601359.

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Li, X., Wang, K. Eigenvalue Estimates on Quaternion-Kähler Manifolds. J Geom Anal 33, 85 (2023). https://doi.org/10.1007/s12220-022-01141-5

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