Upper Bounds for the First Non-zero Steklov Eigenvalue via Anisotropic Mean Curvatures

In this article , we study the eigenvalue problem of the following operator d i v A ∇ f = 0 in M ∂ f ∂ v = p f on ∂ M where M is a compact Riemannian manifold with compact, connected, oriented boundary ∂ M , A denotes a smooth symmetric, positive definite (1,1)-tensor on M . We prove an Reilly-type upper bound for the first non-zero eigenvalue p 1 of this problem via the higher order anisotropic mean curvatures of the boundary ∂ M in R n + 1 . Then, we also prove some pinching theorems for p 1 . In particular, we give some applications of these results to steklov eigenvalue problem.