Asymptotic analysis for approximate harmonic maps from degenerating cylinders and applications to minimal surfaces

TL;DR

This paper analyzes the asymptotic behavior of approximate harmonic maps from degenerating surfaces to Riemannian manifolds, establishing energy identities, describing neck limits, and applying these results to construct minimal cylinders with free boundary.

Contribution

It provides a generalized energy identity, confirms a conjecture on approximate sequences, and introduces a flow method to find minimal cylinders with free boundary.

Findings

Neck regions converge to geodesics or geodesic-like curves.

Energy identities relate the total energy to bubble and neck contributions.

Existence of minimal cylinders with free boundary is established via flow convergence.

Abstract

We investigate the blow-up analysis and quantitative behavior for a sequence of maps $\{u_n\}_{n=1}^\infty$ from degenerating tori $(T^2,g_n)$ or from degenerating cylinders $(S^1\times [0,\pi],g_n)$ with free boundary conditions $u_n(S^1\times \{0,\pi\})\subset K$ to a compact Riemannian manifold $(N,h)$ satisfying $$E(u_n)+\|\tau(u_n,g_n)\|_{L^2}\leq \Lambda<\infty,$$ where $\tau(u_n,g_n)$ is the tension field of $u_n$, $K\subset N$ is a smooth submanifold. We establish generalized energy identities and prove that away from bubbles, the asymptotic limit of the necks are either some geodesics on $N$ or some geodesic-like curves on $K$ where some length formulas are given. This partially confirms a conjecture by Ding-Li-Liu \cite{Ding-Li-Liu} in the sense of approximate sequence case. Moreover, we study an evolution system to seek minimal cylinders in a compact Riemannian manifold with free boundary and with arbitrary codimensions. By studying the convergence of the flow at infinity, we obtain some existence results of minimal cylinders with free boundary. Compared with the closed case in, an interesting new phenomenon here is that the neck may converges to a geodesic-like curve on $K$.