Geometrically and topologically random surfaces in a closed hyperbolic three manifold

Jeremy Kahn; Vladimir Markovic; Ilia Smilga

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arXiv · 2309.02847

Geometrically and topologically random surfaces in a closed hyperbolic three manifold

Jeremy Kahn·Vladimir Markovic·Ilia Smilga

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Abstract

We study the distribution of geometrically and topologically nearly geodesic random surfaces in a closed hyperbolic 3-manifold M. In particular, we describe PSL(2,R) invariant measures on the Grassmann bundle G(M) which arise as limits of random minimal surfaces. It is showed that if M contains at least one totally geodesic subsurface then every topological limiting measure is totally scarring (i.e supported on the totally geodesic locus), while we prove that geometrical limiting measures are never totally scarring.

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Jeremy Kahn, Vladimir Markovic, Ilia Smilga. 2023-09-06. Geometrically and topologically random surfaces in a closed hyperbolic three manifold. https://arxiv.org/abs/2309.02847

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