@article{Xia2024, 
author = {Chao Xia and Changwei Xiong},
title = {Escobar’s Conjecture on a Sharp Lower Bound for the First Nonzero Steklov Eigenvalue},
year = {2024},
journal = {Peking Mathematical Journal},
volume = {7},
number = {2},
pages = {759-778},
keywords = {Steklov eigenvalue, Laplacian eigenvalue, Sharp bound, Nonnegative sectional curvature},
url = {https://www.sciopen.com/article/10.1007/s42543-023-00068-2},
doi = {10.1007/s42543-023-00068-2},
abstract = {It was conjectured by Escobar (J Funct Anal 165:101–116, 1999) that for an ndimensional (n ≥ 3) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by c > 0, the first nonzero Steklov eigenvalue is greater than or equal to c with equality holding only on isometrically Euclidean balls with radius 1/c. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu–Xia’s weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary, as well as a generalized Pohozaev-type identity.}
}