# Uniqueness of minimal surfaces, Jacobi fields, and flat structures

## TL;DR

This paper introduces a new rigidity theorem for associate families of minimal surfaces, particularly Scherk's surfaces, using flat structures and harmonic functions, leading to novel uniqueness results for periodic minimal surfaces.

## Contribution

It develops a new approach combining flat structures and harmonic functions to establish uniqueness theorems for minimal surfaces, extending classical results like Bernstein's theorem.

## Key findings

- Rigidity theorem for associate families of minimal surfaces.
- New uniqueness results for periodic minimal surfaces.
- Application of flat structures and harmonic functions in minimal surface theory.

## Abstract

Inspired by the Finn-Osserman (1964), Chern (1969), do Carmo-Peng (1979) proofs of the Bernstein theorem, which characterizes flat planes as the only entire minimal graphs, we prove a new rigidity theorem for associate families connecting the doubly periodic Scherk graphs and the singly periodic Scherk towers. Our characterization of Scherk's surfaces discovers a new idea from the original Finn-Osserman curvature estimate. Combining two generically independent flat structures introduced by Chern and Ricci, we shall construct geometric harmonic functions on minimal surfaces, and establish that periodic minimal surfaces admit fresh uniqueness results.