About a Proposition on Escobar's paper "The Geometry of the First Non-zero Stekloff Eigenvalue"

Abstract

Let $(M^{2},g_{0})$ be a compact manifold with boundary, and let $g$ and $g_{0}$ be conformally related by $g=e^{2f}g_{0}$. We show that the inequality $$ν_{1}(g)\geq\Big(\max_{x\in\partial M}e^{-f(x)}\Big)ν_{1}(g_{0})$$ stated in Proposition 2 in [1], is only possible when the equality is achieved. In order to achieve such equality, it is required that the function $f$ be constant on $\partial M$, as it is mentioned in Remark 3 also in [1]. Hence, the scope of this inequality is less broad than the one suggested by the Proposition.