Abstract
We establish sharp lower bounds for the first nonzero eigenvalue of the weighted p-Laplacian with \(1< p< \infty \) on a compact Bakry–Émery manifold \((M^n,g,f)\) satisfying \({\text {Ric}}+\nabla ^2 f \ge \kappa \, g\), provided that either \(1<p \le 2\) or \(\kappa \le 0\). For \(1<p \le 2\), we provide a simple proof via the modulus of continuity estimates. The proof for the \(\kappa \le 0\) case is based on a sharp gradient comparison theorem for the eigenfunction together with a careful analysis of the underlying one-dimensional model equation.
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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. Surv. Differ. Geom., vol. 19, pp. 1–47. International Press, Somerville (2015)
Andrews, B., Clutterbuck, J.: Proof of the fundamental gap conjecture. J. Am. Math. Soc. 24(3), 899–916 (2011)
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)
Andrews, B., Ni, L.: Eigenvalue comparison on Bakry-Emery manifolds. Commun. Partial Differ. Equ. 37(11), 2081–2092 (2012)
Andrews, B., Xiong, C.: Gradient estimates via two-point functions for elliptic equations on manifolds. Adv. Math. 349, 1151–1197 (2019)
Bakry, D., Qian, Z.: Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Adv. Math. 155(1), 98–153 (2000)
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
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)
Chen, M., Wang, F.: Application of coupling method to the first eigenvalue on manifold. Prog. Nat. Sci. (English Ed.) 5(2), 227–229 (1995)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77. American Mathematical Society/Science Press Beijing, Providence/ New York (2006)
Dai, X., Seto, S., Wei, G.: Fundamental gap estimate for convex domains on sphere – the case n = 2. Commun. Anal. Geom. (to appear) (2018). arXiv:1803.01115
Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies, vol. 202. Elsevier, Amsterdam (2005)
Escobar, J.: Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate. Commun. Pure Appl. Math. 43(7), 857–883 (1990)
Gong, F., Li, H., Luo, D.: Erratum to: A probabilistic proof of the fundamental gap conjecture via the coupling by reflection [MR3489850]. Potential Anal. 44(3), 443–447 (2016)
Gong, F., Li, H., Luo, D.: A probabilistic proof of the fundamental gap conjecture via the coupling by reflection. Potential Anal. 44(3), 423–442 (2016)
Hamilton, R.S.: The formation of singularities in the Ricci flow. In: Surveys in Differential Geometry, vol. II (Cambridge, MA, 1993), pp. 7–136. International Press, Cambridge (1995)
Hang, F., Wang, X.: A remark on Zhong–Yang’s eigenvalue estimate. Int. Math. Res. Not. IMRN (18), Art. ID rnm064, 9 (2007)
He, C., Wei, G., Zhang, Q.S.: Fundamental gap of convex domains in the spheres. Am. J. Math. 142(4), 1161–1191 (2020)
Koerber, T.: Sharp estimates for the principal eigenvalue of the \(p\)-operator. Calc. Var. Partial Differ. Equ. 57(2), Paper No. 49, 30 (2018)
Kröger, P.: On the spectral gap for compact manifolds. J. Differ. Geom. 36(2), 315–330 (1992)
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)
Li, X.: Moduli of continuity for viscosity solutions. Proc. Am. Math. Soc. 144(4), 1717–1724 (2016)
Li, X.: Modulus of continuity estimates for fully nonlinear parabolic equations (2020). arXiv:2006.16631
Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., vol. XXXVI, pp. 205–239. American Mathematical Society, Providence (1980)
Li, X., Wang, K.: Moduli of continuity for viscosity solutions on manifolds. J. Geom. Anal. 27(1), 557–576 (2017)
Li, X., Wang, K.: Sharp lower bound for the first eigenvalue of the weighted \(p\)-Laplacian II. Math. Res. Lett. (to appear) (2019). arXiv:1911.04596
Li, X., Wang, K.: Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds (2020). arXiv:2010.12792
Li, X., Wang, K., Tu, Y.: On a class of quasilinear operators on smooth metric measure spaces (2020). arXiv:2009.10418
Lichnerowicz, A.: Géométrie des groupes de transformations. III. Dunod, Paris, Travaux et Recherches Mathématiques (1958)
Ling, J., Lu, Z.: Bounds of eigenvalues on Riemannian manifolds. In: Trends in Partial Differential Equations. Adv. Lect. Math. (ALM), vol. 10, pp. 241–264. Int. Press, Somerville (2010)
Matei, A.-M.: First eigenvalue for the \(p\)-Laplace operator. Nonlinear Anal. 39(8, Ser. A: Theory Methods), 1051–1068 (2000)
Naber, A., Valtorta, D.: Sharp estimates on the first eigenvalue of the \(p\)-Laplacian with negative Ricci lower bound. Math. Z. 277(3–4), 867–891 (2014)
Ni, L.: Estimates on the modulus of expansion for vector fields solving nonlinear equations. J. Math. Pures Appl. (9) 99(1), 1–16 (2013)
Ni, L., Zheng, F.: Comparison and vanishing theorems for Kähler manifolds. Calc. Var. Partial Differ. Equ. 57(6), Paper No. 151, 31 (2018)
Ni, L., Zheng, F.: On orthogonal Ricci curvature. In: Advances in Complex Geometry. Contemp. Math., vol. 735, pp. 203–215. American Mathematical Society, Providence (2019)
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)
Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge (1994)
Seto, S., Wang, L., Wei, G.: Sharp fundamental gap estimate on convex domains of sphere. J. Differ. Geom. 112(2), 347–389 (2019)
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51(1), 126–150 (1984)
Tu, Y.: On the lower bound of the principal eigenvalue of a nonlinear operator (2020). arXiv:2008.00185
Valtorta, D.: Sharp estimate on the first eigenvalue of the \(p\)-Laplacian. Nonlinear Anal. 75(13), 4974–4994 (2012)
Wang, L.F.: Eigenvalue estimate for the weighted \(p\)-Laplacian. Ann. Mat. Pura Appl. (4) 191(3), 539–550 (2012)
Wang, Y.-Z., Li, H.-Q.: Lower bound estimates for the first eigenvalue of the weighted \(p\)-Laplacian on smooth metric measure spaces. Differ. Geom. Appl. 45, 23–42 (2016)
Wei, G., Wylie, W.: Comparison geometry for the Bakry-Emery Ricci tensor. J. Differ. Geom. 83(2), 377–405 (2009)
Zhang, Y., Wang, K.: An alternative proof of lower bounds for the first eigenvalue on manifolds. Math. Nachr. 290(16), 2708–2713 (2017)
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)
Acknowledgements
The authors would like to thank Professors Ben Andrews, Zhiqin Lu, Lei Ni, Guofang Wei and Richard Schoen for their interests in this work.
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K. Wang research is supported by NSFC No. 11601359.
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Li, X., Wang, K. Sharp Lower Bound for the First Eigenvalue of the Weighted p-Laplacian I. J Geom Anal 31, 8686–8708 (2021). https://doi.org/10.1007/s12220-021-00613-4
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DOI: https://doi.org/10.1007/s12220-021-00613-4
Keywords
- Eigenvalue estimates
- Weighted p-Laplacian
- Bakry–Émery manifolds
- and modulus of continuity
- Gradient comparison theorem