Mathematical Research Letters

Volume 26 (2019)

Number 3

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

Pages: 903 – 919



Franco Vargas Pallete (Department of Mathematics, University of California at Berkeley)


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.

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

Received 7 April 2016

Accepted 19 June 2018

Published 25 October 2019