Spectral Bernstein theorems for submanifolds in Euclidean spaces

TL;DR

This paper investigates the essential spectrum of submanifolds in Euclidean spaces, establishing conditions under which it equals [0, +inite), based on geometric properties like mean curvature and volume growth.

Contribution

It provides new spectral results for submanifolds under extrinsic geometric conditions, including integrability of the second fundamental form.

Findings

Essential spectrum of certain submanifolds is [0, +inite) under specific curvature conditions.

Finite total mean curvature influences the spectral properties of submanifolds.

The second fundamental form's integrability condition ensures the spectrum starts at zero.

Abstract

In this paper, we consider the essential spectrum of submanifolds in Euclidean spaces under various geometric hypotheses. Our results involve extrinsic conditions such as finite total mean curvature, the convergence of the gradient of the extrinsic distance, and the extrinsic volume growth or the pinching curvature. In particular, we prove that the essential spectrum of a complete non-compact submanifold $M^n$ in a Euclidean space is $[0, +\infty)$ provided the second fundamental form $A$ of $M^n$ satisfies $\|A\|_{L^p} < \infty$, $p>n$.