On the existence of min-max minimal torus

TL;DR

This paper investigates the existence of min-max minimal tori using conformal invariant geometric variational methods, establishing key theorems on uniformization, compactification, and bubbling convergence.

Contribution

It introduces a new existence theorem for min-max minimal tori and develops related uniformization and convergence results using advanced geometric analysis techniques.

Findings

Proves the existence of min-max minimal tori (Theorem 5.1).

Establishes a uniformization result (Proposition 3.1).

Provides bubbling convergence results similar to prior studies.

Abstract

In this paper, we will study the existence problem of minmax minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of minmax minimal torus in Theorem 5.1. Firstly we prove a strong uniformization result(Proposition 3.1) using method of [1]. Then we use this proposition to choose good parametrization for our minmax sequences. We prove a compactification result(Lemma 4.1) similar to that of Colding and Minicozzi [2], and then give bubbling convergence results similar to that of Ding, Li and Liu [7]. In fact, we get an approximating result similar to the classical deformation lemma(Theorem 1.1).