Minimal surfaces with negative curvature in large dimensional spheres

TL;DR

This paper proves the existence of closed minimal surfaces with negative curvature in high-dimensional spheres, extending Yau's question and employing advanced geometric techniques.

Contribution

It introduces a novel method to construct almost hyperbolic minimal surfaces with negative curvature in large spheres, answering a longstanding question.

Findings

Existence of negatively curved minimal surfaces in large spheres confirmed

Method extends to surfaces with large automorphism groups

Constructs almost hyperbolic minimal surfaces in high dimensions

Abstract

In this note, we answer positively a question of Yau by proving the existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension. The proof follows the strategy of Song, applying it to closed Riemann surfaces with large automorphism groups, and obtaining almost hyperbolic minimal surfaces.