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- W4313305482 abstract "In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a higher-dimensional analogue of the geodesic flow when the ambient manifold is negatively curved. In this paper, following the recent work [5] of Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups in terms of areas of $k$-surfaces. We determine a lower bound, and we prove rigidity when this bound is achieved. Our work differs from that of [5] in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5] breaks down in the present case, we require a different approach. Following ideas outlined by Labourie in [19], we prove rigidity by solving a general foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build on Labourie's theory of $k$-surface dynamics, and propose a number of new constructions, conjectures and questions." @default.
- W4313305482 created "2023-01-06" @default.
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- W4313305482 date "2022-12-27" @default.
- W4313305482 modified "2023-09-23" @default.
- W4313305482 title "Foliated Plateau problems and asymptotic counting of surface subgroups" @default.
- W4313305482 doi "https://doi.org/10.48550/arxiv.2212.13604" @default.
- W4313305482 hasPublicationYear "2022" @default.
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