Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A necessary and sufficient condition for Ricci shrinkers to have
  positive AVR   In this short note we observe that a recent result of C.-W. Chen meshes well
with earlier work of H.-D. Cao and D.-T. Zhou, O. Munteanu, J. Carrillo and L.
Ni, and S.-J. Zhang to give a necessary and sufficient condition for complete
noncompact shrinking gradient Ricci solitons to have positive asymptotic volume
ratio.

21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph