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

Leoncio Rodriguez Quiñones

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.

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

Abstract

Let be a compact manifold with boundary, and let and be conformally related by . We show that the inequality 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 be constant on , 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.
Paper Structure (4 sections, 1 theorem, 17 equations)

This paper contains 4 sections, 1 theorem, 17 equations.

Table of Contents

  1. Discussion
  2. Declarations
  3. Funding
  4. Acknowledgment

Key Result

Proposition 1

Inequality $Q_{g}(\phi) \geq(\max_{\partial M}e^{-f})Q_{g_{0}}(\phi)$ only holds for the equality case, and such equality occurs if $f$ is constant on $\partial M$. If $f$ is not constant, equality is not guaranteed.

Theorems & Definitions (2)

  • Proposition 1
  • proof