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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.18630 |
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| _version_ | 1866912601337757696 |
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| author | Chu, Adrian Chun-Pong Stern, Daniel |
| author_facet | Chu, Adrian Chun-Pong Stern, Daniel |
| contents | Twenty years ago, N. Kapouleas introduced a singular perturbation construction known as "doubling", which produces sequences of high-genus minimal surfaces converging to a given minimal surface with multiplicity two. Doubling constructions have since been implemented successfully in several settings, with deep work of Kapouleas-McGrath reducing their existence theory to the problem of finding suitable families of ansatz data on the initial minimal surface.
In this paper, we introduce a variational approach to the existence of minimal doublings, relating the Kapouleas-McGrath construction to the study of nondegenerate critical points for a Coulomb-type interaction energy. By analyzing the minimizers of this energy, we prove that, in a generic closed 3-manifold, every two-sided, embedded minimal surface of index one admits a sequence of minimal doublings. As a corollary, we find that a generic 3-manifold contains an infinite sequence of embedded minimal surfaces with bounded area and arbitrarily large genus, whose geometry can be described with some precision. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18630 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal surface doublings and electrostatics for Schrödinger operators Chu, Adrian Chun-Pong Stern, Daniel Differential Geometry Analysis of PDEs Twenty years ago, N. Kapouleas introduced a singular perturbation construction known as "doubling", which produces sequences of high-genus minimal surfaces converging to a given minimal surface with multiplicity two. Doubling constructions have since been implemented successfully in several settings, with deep work of Kapouleas-McGrath reducing their existence theory to the problem of finding suitable families of ansatz data on the initial minimal surface. In this paper, we introduce a variational approach to the existence of minimal doublings, relating the Kapouleas-McGrath construction to the study of nondegenerate critical points for a Coulomb-type interaction energy. By analyzing the minimizers of this energy, we prove that, in a generic closed 3-manifold, every two-sided, embedded minimal surface of index one admits a sequence of minimal doublings. As a corollary, we find that a generic 3-manifold contains an infinite sequence of embedded minimal surfaces with bounded area and arbitrarily large genus, whose geometry can be described with some precision. |
| title | Minimal surface doublings and electrostatics for Schrödinger operators |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2509.18630 |