Lichnerowicz Type Estimate for the $p$-Laplacian Under Weighted Integral Curvature Bounds

Year 2024, Volume: 17 Issue: 1, 245 - 251, 23.04.2024

Shoo Seto

https://doi.org/10.36890/iejg.1436665

Abstract

In this short note, we prove a quantitative lower bound in terms of the dimension and curvature,
known as a Lichnerowicz-type estimate, for the first eigenvalue of the p-Laplacian on Riemannian
manifolds with a bound on the integral norm of the Bakry-Émery curvature.

Keywords

$p$-Laplacian, Bakry-Emery curvature, integral curvature

Thanks

Thank you for your consideration

References

• [1] Aubry, E.: Finiteness of π1 and geometric inequalities in almost positive Ricci curvature. Ann. Sci. École Norm. Sup. 40 (4), 675–695 (2007).
• [2] Bakry D., Émery, M.: Diffusions hypercontractives. Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 177–206 (1985).
• [3] Gallot, S.: Isoperimetric inequalities based on integral norms of Ricci curvature. Astérisque 157–158, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), 191–216 (1988).
• [4] Li, F., Wu, J.-Y., Zheng, Y.: Myers’ type theorem for integral Bakry-Émery Ricci tensor bounds. Results Math. 76 (1), Paper No. 32, (2021).
• [5] Petersen, P., Wei, G., Relative volume comparison with integral curvature bounds. Geom. Funct. Anal. 7 (6), 1031–1045 (1997).
• [6] Ramos Olivé, X., Seto, S. Gradient Estimates of a nonlinear parabolic equation under integral Bakry-Émery Ricci condition, Preprint (2024).
• [7] Seto, S., Wei, G. First eigenvalue of the p-Laplacian under integral curvature condition. Nonlinear Anal. 163, 60–70 (2017).
• [8] Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equation. J. Differential Equations 51 (1), 126–150 (1984).
• [9] Wang, Y.-Z., Li, H.-Q.Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces. Differential Geom. Appl. 45, 23–42 (2016).
• [10] Wang, L., Wei, G. Local Sobolev constant estimate for integral Bakry-Émery Ricci curvature. Pacific J. Math. 300 (1), 233–256 (2019).
• [11] Wei, G., Wylie, W. Comparison geometry for the Bakry-Émery Rici tensor. J. Differential Geom. 83 (2), 377–405 (2009).
• [12] Wu, J.-Y. Comparison geometry for integral Bakry-Émery Ricci tensor bounds. J. Geom. Anal. 29 (1), 828–867 (2019).