Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces

TL;DR

This paper investigates the asymptotic growth rates and rigidity properties of high energy $k$-surfaces in negatively curved 3-manifolds, establishing bounds and conditions for spectral asymptotics related to curvature.

Contribution

It provides new rigidity theorems and bounds for the energy spectrum of $k$-surfaces, linking spectral asymptotics to the ambient space's curvature.

Findings

Rigid upper bounds for growth rates of quasi-Fuchsian $k$-surfaces.

Spectral asymptotics occur only in constant curvature spaces.

Domination and rigidity theorems for the marked energy spectrum.

Abstract

Labourie raised the question of determining the possible asymptotics for the growth rate of compact $k$-surfaces, counted according to energy, in negatively curved $3$-manifolds, indicating the possibility of a theory of thermodynamical formalism for this class of surfaces. Motivated by this question and by analogous results for the geodesic flow, we prove a number of results concerning the asymptotic behavior of high energy $k$-surfaces, especially in relation to the curvature of the ambient space. First, we determine a rigid upper bound for the growth rate of quasi-Fuchsian $k$-surfaces, counted according to energy, and with asymptotically round limit set, subject to a lower bound on the sectional curvature of the ambient space. We also study the marked energy spectrum for $k$-surfaces, proving a number of domination and rigidity theorems in this context. Finally, we show that the marked area and energy spectra for $k$-surfaces in $3$-dimensional manifolds of negative curvature are asymptotic if and only if the sectional curvature is constant.