Unstable minimal surfaces in symmetric spaces of non-compact type

2022/08/09 by Sagman, Nathaniel, Smillie, Peter · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2208.04885

Abstract

We prove that if Σ is a closed surface of genus at least 3 and G is a split real semisimple Lie group of rank at least 3 acting faithfully by isometries on a symmetric space N, then there exists a Hitchin representation ρ:π₁(Σ)→ G and a ρ-equivariant unstable minimal map from the universal cover of Σ to N. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking G=PSL(n,ℝ), n≥ 4, this disproves the Labourie conjecture.

Cited by

• Expanding Ricci solitons and Higgs bundles

Related

• Cyclic surfaces and Hitchin components in rank 2
• Minimal surfaces for Hitchin representations
• High energy harmonic maps and degeneration of minimal surfaces
• Unstable minimal surfaces in ℝⁿ and in products of hyperbolic surfaces
• Hitchin harmonic maps are immersions