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 $${\nu }_1(g)\ge \left(\underset{x\in \partial M}{\max }{e}^{-f(x)}\right){\nu }_1(g_0)$$ 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.