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Continuity of the renormalized volume under geometric limits
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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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Cited by 1 Pith paper
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Behaviour of the Schwarzian derivative on long complex projective tubes
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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