The first Stekloff eigenvalue in weighted Riemannian manifolds

In study of eigenvalue problems, a classical problem is the Stekloff eigenvalue problem. There are many estimates of the first non- zero Stekloff eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997)". In this paper we are interested to study this problem in weighted context. Various estimates are obtained, including a sharp estimate on surfaces, similar to that Escobar obtained.