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Continuity of the renormalized volume under geometric limits

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arxiv 1605.07986 v1 pith:HA5DOFYM submitted 2016-05-25 math.DG

classification math.DG
keywords classgeometricallyrenormalizedvolumeconformalfinitegeodesichyperbolic
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abstract

We extend the concept of renormalized volume for geometrically finite hyperbolic $3$-manifolds, and show that is continuous for geometrically convergent sequences of hyperbolic structures over an acylindrical 3-manifold $M$ with geometrically finite limit. This allows us to show that the renormalized volume attains its minimum (in terms of the conformal class at $\partial M = S$) at the geodesic class, the conformal class for which the boundary of the convex core is totally geodesic.

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  1. Behaviour of the Schwarzian derivative on long complex projective tubes

    math.DG 2025-02 accept novelty 7.0 of 10

    On long complex projective tubes, the Schwarzian derivative is asymptotically (1/2z^2)(1+4π^2/ℓ^2) dz^2 up to exponentially small errors, giving control of renormalized volume under earthquakes and grafting.

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