Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Renormalized volume on the Teichm\"uller space of punctured surfaces   We define and study the renormalized volume for geometrically finite
hyperbolic $3$-manifolds, including with rank-$1$ cusps. We prove a variation
formula, and show that for certain families of convex co-compact hyperbolic
metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric $g_0$
with rank-$1$ cusps, the renormalized volume converges to the renormalized
volume of the limiting metric.

30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph