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标题: "Curvature at Infinity Governs the Topology of Complete Non-Compact Surfaces Admitting Schrödinger Operators of Finite Index"
原文链接: "https://arxiv.org/abs/2608.10518"
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## Mathematics > Differential Geometry

## Title:Curvature at Infinity Governs the Topology of Complete Non-Compact Surfaces Admitting Schrödinger Operators of Finite Index

Authors:[Hideaki Harumoto](https://arxiv.org/search/math?searchtype=author&query=Harumoto,+H), [Kei Kondo](https://arxiv.org/search/math?searchtype=author&query=Kondo,+K)

[View PDF](https://arxiv.org/pdf/2608.10518) [HTML (experimental)](https://arxiv.org/html/2608.10518v1)

> Abstract:In this article, we investigate the global topology of a complete non-compact Riemannian $2$-manifold $\Sigma$ admitting a Schrödinger operator with non-negative potential and finite Morse index. While classical results of Fischer-Colbrie classify such manifolds under the assumption of vanishing index or geometric stability as immersed minimal surfaces in a Riemannian $3$-manifold, we show that the curvature at infinity $\lambda_\infty^*(\Sigma)$ of the Fischer-Colbrie metric $g^*$ ---a complete conformal metric determined by a positive function furnished by Fischer-Colbrie's theorem---governs the global topology and geometric rigidity of $\Sigma$ without imposing either assumption.  
> More precisely, we derive a fundamental identity relating $\lambda_\infty^*(\Sigma)$ to the area growth of $(\Sigma,g^*)$, show that all critical points of the distance function $d_p^*$ from a fixed base point $p$ are confined to a bounded region, and, as a corollary, obtain a quantitative bound for the number of ends. We further distinguish two complementary geometric viewpoints. On the one hand, a quantitative condition on $\lambda_\infty^*(\Sigma)$ forces $\Sigma$ to be diffeomorphic to the Euclidean plane $\mathbb{R}^2$. On the other hand, when $\Sigma$ has exactly one end, another condition on $\lambda_\infty^*(\Sigma)$ guarantees that every Busemann function on $(\Sigma,g^*)$ is an exhaustion. By clarifying the relationship between these two regimes---the critical-point structure of distance functions relative to a base point and the global behavior of Busemann functions at infinity---we exhibit two complementary manifestations of how $\lambda_\infty^*(\Sigma)$ controls the global geometry and topology of $\Sigma$.

| Comments: |  |
| --- | --- |
| Subjects: | Differential Geometry (math.DG); Analysis of PDEs (math.AP) |
| MSC classes: | 53C20, 58J05, 35J10, 35P15 |
| Cite as: | [arXiv:2608.10518](https://arxiv.org/abs/2608.10518) \[math.DG\] |
|  | (or [arXiv:2608.10518v1](https://arxiv.org/abs/2608.10518v1) \[math.DG\] for this version) |
|  | [https://doi.org/10.48550/arXiv.2608.10518](https://doi.org/10.48550/arXiv.2608.10518) |

## Submission history

From: Kei Kondo \[[view email](https://arxiv.org/show-email/8fcc9ad4/2608.10518)\]  
**\[v1\]** Tue, 11 Aug 2026 05:46:12 UTC (24 KB)

[Which authors of this paper are endorsers?](https://arxiv.org/auth/show-endorsers/2608.10518) | Disable MathJax ([What is MathJax?](https://info.arxiv.org/help/mathjax.html))
