Geometrically and topologically random surfaces in a closed hyperbolic three manifold

Ilia Smilga; Jeremy Kahn; Vladimir Markovic

Geometrically and topologically random surfaces in a closed hyperbolic three manifold

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arxiv 2309.02847 v1 pith:XATPUAB6 submitted 2023-09-06 math.GT math.DG

Jeremy Kahn , Vladimir Markovic , Ilia Smilga This is my paper · ORCID

classification math.GTmath.DG

keywords totallygeodesicrandomsurfacesclosedgeometricallyhyperboliclimiting

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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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