Local convexity of renormalized volume for $\textrm{rank-1}$ cusped manifolds

We study the critical points of the renormalized volume for acylindrical geometrically finite hyperbolic $3$-manifolds that include $\textrm{rank-1}$ cusps, and show that the renormalized volume is locally convex around these critical points. We give a modified definition of the renormalized volume that is additive under gluing, and study some local properties.

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The author’s research was partially supported by NSF grant DMS-1406301.

Received 7 April 2016

Accepted 19 June 2018

Published 25 October 2019