On the first Neumann eigenvalue for critical points of a weighted area functional with asymptotically flat ends

TL;DR

This paper establishes a lower bound for the first Neumann eigenvalue of the drift Laplacian on certain minimal surfaces with asymptotically flat ends, linking spectral properties to geometric and topological features.

Contribution

It provides a novel lower bound for the Neumann eigenvalue on weighted minimal surfaces with asymptotically flat ends, connecting spectral estimates to surface topology.

Findings

Lower bound for the first Neumann eigenvalue of the drift Laplacian.

Control on the topology of surfaces with finite total curvature.

Relation between eigenvalues and Poincaré constants.

Abstract

In the following work, we obtain a lower bound for the first Neumann eingevalue of the drift Laplacian $\Delta^{\varphi}$ for a family of properly embedded $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$ with concave function $\varphi$ and asymptotically flat ends. As application, we obtain a control on the topology for that surfaces with finite total curvature. The strategy will consists in an integration of the Bouchner's formula by the works of A. Lichnerowicz and S. Brendle, R. Tsiamis and relate said eigenvalue with the Poincare's constant.