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

Sébastien Alvarez; Ben Lowe; Graham Andrew Smith

doi:10.5802/jep.309

Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces
[Rigidité du spectre marqué des énergies de la métrique hyperbolique et entropie pour les $k$-surfaces]

Sébastien Alvarez ¹ ; Ben Lowe ² ; Graham Andrew Smith ³

¹ CMAT, Facultad de Ciencias, Universidad de la República, & IRL-IFUMI (CNRS), Igua 4225 esq. Mataojo. Montevideo, Uruguay
² Department of Mathematics, University of Chicago, Chicago IL 60637, USA
³ Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro 225453-900, Brazil

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1197-1227.

Abstract (VO)
Résumé

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.

Labourie a soulevé la question de déterminer les comportements asymptotiques possibles pour le taux de croissance des $k$-surfaces compactes, comptées selon leur énergie, dans les $3$-variétés de courbure négative, en suggérant la possibilité d’une théorie du formalisme thermodynamique pour cette classe de surfaces. Motivés par cette question et par des résultats analogues pour le flot géodésique, nous démontrons plusieurs résultats concernant le comportement asymptotique des $k$-surfaces de grande énergie, en particulier en lien avec la courbure de l’espace ambiant.

Premièrement, nous établissons une borne supérieure rigide pour le taux de croissance des $k$-surfaces quasi-fuchsiennes, comptées selon leur énergie et ayant un ensemble limite asymptotiquement circulaire, sujet à une borne inférieure pour la courbure sectionnelle de l’espace ambiant. Nous étudions également le spectre marqué des énergies des $k$-surfaces, en prouvant plusieurs théorèmes de domination et de rigidité dans ce contexte. Enfin, nous montrons que les spectres marqués des aires et des énergies des $k$-surfaces dans les $3$-variétés de courbure négative sont asymptotiques si et seulement si la courbure sectionnelle est constante.

Reçu le : 2024-12-19
Accepté le : 2025-08-14
Publié le : 2025-08-19

DOI : 10.5802/jep.309

Classification : 37C85, 57M50, 53C42
Keywords: Geometric rigidity, equidistribution, surfaces of constant curvature, homogeneous actions of Lie groups
Mots-clés : Rigidité géométrique, équidistribution, surfaces de courbure constante, actions homogènes de groupes de Lie

Affiliations des auteurs :

Sébastien Alvarez ¹ ; Ben Lowe ² ; Graham Andrew Smith ³

¹ CMAT, Facultad de Ciencias, Universidad de la República, & IRL-IFUMI (CNRS), Igua 4225 esq. Mataojo. Montevideo, Uruguay
² Department of Mathematics, University of Chicago, Chicago IL 60637, USA
³ Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro 225453-900, Brazil

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Sébastien Alvarez; Ben Lowe; Graham Andrew Smith. Rigidity of the hyperbolic marked energy spectrum and entropy for $k$-surfaces. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1197-1227. doi : 10.5802/jep.309. https://jep.centre-mersenne.org/articles/10.5802/jep.309/

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