Minimal surfaces with negative curvature in large dimensional spheres

TL;DR

The authors construct closed minimal surfaces with negative induced curvature in round spheres of arbitrarily large dimension, answering Yau's question. They refine Song's equivariant harmonic-map strategy by replacing the 3-holed sphere with an orbifold whose Teichmüller space is a point and by using induced representations to lift finite-energy data from a torsion-free subgroup to the ambient group. The main result yields a sequence of surfaces $\Sigma_n\subset \mathbb{S}^n$ with curvature $\kappa_n$ approaching $-8$, and a correspondence with a fixed Riemann surface $X$ via $X=\Gamma_n\backslash\Sigma_n$, showing asymptotically Bryant’s obstruction does not hold. The proof hinges on energy convergence results for equivariant harmonic maps, the induced-representation construction, and Sacks–Uhlenbeck regularity, culminating in negatively curved, embedded minimal immersions in high-dimensional spheres. This advances the understanding of negative curvature in minimal surface theory and provides a framework for constructing such surfaces using representation theory and orbifold techniques.

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.

Theorems & Definitions (27)

• Corollary 1.1
• Remark 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Theorem 2.5: Sacks--Uhlenbeck, Song
• proof : Proof of the extended of Sacks--Uhlenbeck Theorem \ref{['thm:SacksUhlenbeck2']}
• Theorem 2.6: Louder--Magee
• Theorem 2.7: Song's Convergence Theorem
• Theorem 2.8: Song's rigidity theorem