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Convergence of the gradient flow of renormalized volume to convex cores with totally geodesic boundary

https://doi.org/10.1112/S0010437X2300708X

Bridgeman, Martin ; Bromberg, Kenneth ; Vargas Pallete, Franco ( April 2023 , Compositio Mathematica)

We consider the Weil–Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the unique structure $M_{\rm geod}$ with minimum convex core volume.
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Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

https://doi.org/10.1017/fms.2023.64

Liu, Beibei ; Vargas Pallete, Franco ( January 2023 , Forum of Mathematics, Sigma)

Abstract
We show convergence of small eigenvalues for geometrically finite hyperbolicn-manifolds under strong limits. For a class of convergent convex sets in a strongly convergent sequence of Kleinian groups, we use the spectral gap of the limit manifold and the exponentially mixing property of the geodesic flow along the strongly convergent sequence to find asymptotically uniform counting formulas for the number of orthogeodesics between the convex sets. In particular, this provides asymptotically uniform counting formulas (with respect to length) for orthogeodesics between converging Margulis tubes, geodesic loops based at converging basepoints, and primitive closed geodesics.

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Minimal area surfaces and fibered hyperbolic 3-manifolds

https://doi.org/10.1090/proc/15998

Farre, James ; Vargas Pallete, Franco ( November 2022 , Proceedings of the American Mathematical Society)

By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic 3 3 -manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic 3 3 -manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
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The length of the shortest closed geodesic on positively curved 2-spheres

https://doi.org/10.1007/s00209-021-02875-8

Adelstein, Ian ; Vargas Pallete, Franco ( March 2022 , Mathematische Zeitschrift)

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