On the essential spectra of submanifolds in the hyperbolic space

TL;DR

This paper explores how the asymptotic geometry of submanifolds in hyperbolic space influences their boundary regularity and spectral properties, especially focusing on minimal submanifolds and their essential spectrum of the Laplacian.

Contribution

It provides new insights into the relationship between asymptotic geometry and spectral theory of submanifolds in hyperbolic space, including conditions for orthogonal boundary meeting and spectrum computation.

Findings

Characterized when minimal submanifolds meet the ideal boundary orthogonally.

Computed the essential spectrum of the Laplace operator for asymptotically minimal submanifolds.

Revisited and extended existing results on asymptotic geometry and spectral properties.

Abstract

We study relationships between asymptotic geometry of submanifolds in the hyperbolic space and their regularity properties near the ideal boundary, revisiting some of the related results in the literature. In particular, we discuss hypotheses when minimal submanifolds meet the ideal boundary orthogonally, and compute the essential spectrum of the Laplace operator on submanifolds that are asymptotically close to minimal submanifolds.