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Explore limit measures for topologically and geometrically random surfaces in this mathematical lecture from the ZhengTong Chern-Weil Symposium. Examine how immersed surfaces in Riemannian manifolds induce probability measures on the space of two-planes in the tangent bundle, with particular focus on hyperbolic 3-manifolds. Discover the behavior of sequences of surfaces with principal curvatures approaching zero and learn how their weak* limits form convex combinations of Liouville measures and measures from immersed totally geodesic surfaces. Investigate two distinct approaches to generating random nearly geodesic surfaces: one constrained by genus bounds and another by area bounds. Understand the fundamental differences in their limiting behavior, where genus-bounded sequences yield measures exclusively from totally geodesic surfaces (when present), while area-bounded sequences must contain equidistributed portions. This research presentation, delivered by Jeremy Kahn from Brown University, represents collaborative work with V. Markovic and I. Smilga, offering insights into the intersection of differential geometry, probability theory, and hyperbolic geometry.
