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Topology/Geometry Seminar

Distribution of minimal surfaces in compact hyperbolic 3-manifolds

Ilia Smilga (Oxford)

Location:  Online (click on the zoom link)
Date & time: Tuesday, 28 November 2023 at 4:00PM - 5:00PM

In a classical work, Bowen and Margulis proved the equidistribution of closed geodesics in any hyperbolic manifold. Together with Jeremy Kahn and Vladimir Markovi?, we asked ourselves what happens in a three-manifold if we replace curves by surfaces. The natural analog of a closed geodesic is then a minimal surface, as totally geodesic surfaces exist only very rarely. Nevertheless, it still makes sense (for various reasons, in particular to ensure uniqueness of the minimal representative) to restrict our attention to surfaces that are almost totally geodesic.

The statistics of these surfaces then depend very strongly on how we order them: by genus, or by area. If we focus on surfaces whose *area* tends to infinity, we conjecture that they do indeed equidistribute; we proved a partial result in this direction. If, however, we focus on surfaces whose *genus* tends to infinity, the situation is completely opposite: we proved that they then accumulate onto the totally geodesic surfaces of the manifold (if there are any).

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