2024/02/15 by Song, Antoine · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2402.10287
Let S be a punctured Riemann surface with Euler characteristic χ(S)<0. For any unitary representation ρ: π1(S) → U(N), we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic maps from the universal cover of S to the unit sphere in ℂN. Our main result is that if a sequence of unitary representations ρj strongly converges, then their renormalized energies converge to \fracπ4|χ(S)| and the shape of their harmonic representatives converges to a unique rescaled hyperbolic metric. Combining this statement with examples of strongly converging representations provided by random matrix theory, we derive the following applications. (1) If π1(S) is a free group, then for a random ρ: π1(S) → U(N), the shape of its harmonic representatives concentrates around a rescaled hyperbolic metric with high probability as N→ ∞. (2) For any closed hyperbolic surface, a finite covering admits a harmonic immersion into some Euclidean unit sphere, which is almost isometric after rescaling. (3) There are closed, branched, minimal surfaces \mathfrakSj in some Euclidean unit spheres such that \mathfrakSj Benjamini-Schramm converges to a rescaled hyperbolic plane as j→ ∞, and the Gaussian curvature Kj of \mathfrakSj satisfies limj→ ∞ \frac1Area(\mathfrakSj)∫_\mathfrakSj |Kj+8|=0.