<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$M^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> in a Euclidean space is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0, +\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> provided the second fundamental form <i>A</i> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Vert A\Vert _{L^p} < \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>A</mi> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> </msub> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p>n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Spectral Bernstein theorems for submanifolds in Euclidean spaces

期刊论文
发表日期：

2026年5月20日

查看全文

Yuxin Dong,
Hezi Lin,
Wei Zhang

摘要

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 $M^n$ $M^{n}$ in a Euclidean space is $[0, +\infty )$ $[0, + \infty)$ provided the second fundamental form A of $M^n$ $M^{n}$ satisfies $\Vert A\Vert _{L^p} < \infty $ ${‖ A ‖}_{L^{p}} < \infty$ , $p>n$ $p > n$ .