Abstract
In this paper, we investigate the eigenvalues of the drifting Laplacian on the bounded domain in Hardamard manifolds. By using upper half-plane model, we establish a universal inequality for the drifting Laplacian with a specific class of potential functions on the hyperbolic space, which can be viewed as a rigidity result associated with such a class of functions. Furthermore, we consider general Hardamard manifolds with pinching condition of sectional curvature, and obtain an eigenvalue inequality without the condition of Bakry-Émery curvature. As a by-product, we obtain a universal bound for the case of radial drifting Laplacian satisfying certain growth condition along with the radial direction.
Acknowledgment
The partial work is completed in Beijing International Center for Mathematical Research (BICMR), therefore, the second author would like to express his sincere gratitude to BICMR for her enthusiastic help and kindly support. The second author is so grateful to professor Qing-Ming Cheng for his continuous encouragement. We also thank professor Daguang Chen for his helpful discussion on various stages of the work. Many thanks go to the anonymous referee for his or her helpful comments and suggestions. The second author (corresponding author) was supported by the National Natural Science Foundation of China (Grant Nos. 11861036 and 11826213) and the Natural Science Foundation of Jiangxi Province (Grant No. 20171ACB21023).
Citation
Xin Xiong. Lingzhong Zeng. Huihui Zhu. "Estimates for the higher eigenvalues of the drifting Laplacian on Hadamard manifolds." Kodai Math. J. 45 (1) 143 - 156, March 2022. https://doi.org/10.2996/kmj/kmj45109
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