Spectral Bernstein theorems for submanifolds in Euclidean spaces

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

• The paper establishes that under integral bounds on mean curvature and the second fundamental form, the essential spectrum of a submanifold is [0, +∞).
• It leverages mollification of distance functions, volume comparison estimates, and sharp integral inequalities to connect extrinsic geometric conditions with spectral properties.
• The results generalize classical Bernstein-type theorems by removing minimality restrictions and providing a framework for analyzing spectral geometry under weaker curvature conditions.

Spectral Bernstein Theorems for Submanifolds in Euclidean Spaces

Overview

The paper "Spectral Bernstein theorems for submanifolds in Euclidean spaces" (2605.21370) rigorously investigates conditions under which the essential spectrum of the Laplace operator on a complete, non-compact submanifold $M^n$ of Euclidean space $\mathbb{R}^m$ coincides with that of $\mathbb{R}^m$, i.e., $[0, +\infty)$. The analysis relies on extrinsic geometric hypotheses, primarily integral bounds of the second fundamental form and mean curvature, as well as volume growth and asymptotic geometric pinching conditions. This extends classical spectral theory in geometric analysis, particularly in the context of Bernstein-type results.

Main Results

Essential Spectrum Characterization

The central theorems establish that a variety of integral and asymptotic extrinsic conditions suffice to guarantee the $L^2$ essential spectrum of $M^n$ is $[0, +\infty)$:

• Finite Total Mean Curvature and Sub-exponential Volume Growth: If $M^n$ is properly immersed and the $L^n$ norm of its mean curvature vector $H$ is finite, with sub-exponential extrinsic volume growth, then $\mathbb{R}^m$0 (Theorem 1.1).
• Integral Decay of Traceless Second Fundamental Form: If $\mathbb{R}^m$1 is asymptotically umbilical in the sense that $\mathbb{R}^m$2, $\mathbb{R}^m$3, then $\mathbb{R}^m$4 (Theorem 1.2).
• $\mathbb{R}^m$5-Bounded Second Fundamental Form: If $\mathbb{R}^m$6, $\mathbb{R}^m$7, then the essential spectrum is trivial (Corollary 1.1, 4.1).

These results are derived using mollifications of distance functions, volume comparison estimates, and sharp integral inequalities relating curvature and volume growth.

Techniques and Proof Structure

• Charalambous-Lu Criterion: The authors utilize an extension of Weyl's criterion (Lemma 2.3), constructing explicit approximate eigenfunctions whose supports escape compact sets, ensuring the spectral measure covers $\mathbb{R}^m$8.
• Integral and Volume Estimates: A sequence of lemmas grounds the connection between volume growth and curvature decay (Sections 3–4), leveraging results from Sturm, Petersen-Wei, and others to control asymptotic behaviours of geometric quantities.

Comparison with Existing Literature

The analysis situates itself among classical and modern spectral geometry results:

• Donnelly and Kumura previously related spectrum to curvature conditions at infinity for various ambient spaces.
• Prior studies emphasized minimality or boundedness; here, minimality is removed as a restriction, broadening applicability.
• The integral conditions employed can be seen as generalizations from pointwise pinching and bounded geometry, offering greater flexibility in describing geometric decay.

Implications

Theoretical Impact

• Generalization of Bernstein-Type Theorems: These results significantly expand the class of submanifolds for which the essential spectrum is trivial, relaxing both minimality and boundedness requirements.
• Spectral Geometry–Extrinsic Geometry Bridge: The developed criteria firmly connect spectral properties to integral extrinsic geometric invariants, e.g., $\mathbb{R}^m$9 norms of the second fundamental form, rather than intrinsic curvature bounds alone.
• Methodological Advancement: The use of smoothing techniques, integral decay arguments, and volume comparison sets a robust framework for future exploration of spectra under weaker geometric conditions.

Practical Applications

• Spectral Analysis in Geometric PDEs: Understanding the essential spectrum informs the solvability and behaviour of evolution equations and quantum models on manifolds with prescribed extrinsic curvature decay.
• Geometry of Physical Models: Results potentially inform settings where manifolds are modeled as submanifolds in higher-dimensional ambient spaces (e.g., theoretical physics, geometric flows).

Numerical Results and Bold Claims

The paper asserts, without restriction to minimal submanifolds, that the $\mathbb{R}^m$0 essential spectrum under the integral conditions described is always $\mathbb{R}^m$1. No explicit numerical bounds for eigenvalues are included, but the criteria guarantee the absence of discrete spectrum and spectral gaps under the indicated geometric hypotheses.

Speculation on Future Developments

Future work could address:

• Extension to Ambient Spaces of Non-Euclidean Geometry: Whether analogues hold for ambient spaces with variable curvature or other geometric constraints.
• Weaker Curvature Conditions: Seeking even less restrictive integrability or asymptotic decay requirements.
• Spectral Stability and Dynamical Properties: Analyses of the spectral stability under deformations or geometric flows, particularly in the sub-exponential and polynomial volume growth regimes.

Conclusion

The paper rigorously characterizes when submanifolds in Euclidean space possess a trivial essential spectrum, using integral geometric conditions on the second fundamental form and mean curvature, together with volume growth constraints. This advances spectral geometry by connecting extrinsic invariants with spectral completeness, laying groundwork for broader investigations into spectral properties for a rich variety of submanifolds immersed in Euclidean and potentially non-Euclidean spaces (2605.21370).