极小极大理论与具有给定亏格的极小曲面
微分几何
2026-05-01 v3 偏微分方程分析
几何拓扑
摘要
我们建立了一个一般的极小极大型定理，在具有正 Ricci 曲率的三维流形中产生具有给定亏格的极小曲面。一个重要的中间步骤是证明，在具有正 Ricci 曲率的通用度量中，任何光滑嵌入曲面族（可能具有有限个奇点）都可以形变为某个拓扑最优族。本文的结果对于我们通过拓扑方法在三维球面中构造多个具有给定亏格的极小曲面的计划至关重要。
引用
@article{arxiv.2507.23239,
title = {Min-max theory and minimal surfaces with prescribed genus},
author = {Adrian Chun-Pong Chu and Yangyang Li and Zhihan Wang},
journal= {arXiv preprint arXiv:2507.23239},
year = {2026}
}
备注
The v1 manuscript has been split into two parts: the construction of a topologically optimal family is contained in v2, while the existence result for genus 2 minimal surfaces is strengthened and superseded by arXiv:2601.01736. v3: expanded the discussion of the main ideas in section 1.1