In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N \times \mathbb R$, submanifolds with bounded $\varpi$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p$-Laplacian.
This work was supported in part by the NSF of China (Grant Nos. 11401131 and 11801496), the Fok Ying-Tung Education Foundation (China), and Key Laboratory of Applied Mathematics of Hubei Province (Hubei University).
"Eigenvalue estimates for submanifolds in Hadamard manifolds and product manifolds $N \times \mathbb R$." Hiroshima Math. J. 50 (1) 17 - 42, March 2020. https://doi.org/10.32917/hmj/1583550013