Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the classification of ancient solutions to curvature flows on the
  sphere   We consider the evolution of hypersurfaces on the unit sphere
$\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the
notion of `quasi-ancient' solutions for flows that do not admit non-trivial,
convex, ancient solutions. Such solutions are somewhat analogous to ancient
solutions for flows such as the mean curvature flow, or 1-homogeneous flows.
The techniques presented here allow us to prove that any convex, quasi-ancient
solution of a curvature flow which satisfies a backwards in time uniform bound
on mean curvature must be stationary or a family of shrinking geodesic spheres.
The main tools are geometric, employing the maximum principle, a rigidity
result in the sphere and an Alexandrov reflection argument. We emphasize that
no homogeneity or convexity/concavity restrictions are placed on the speed,
though we do also offer a short classification proof for several such
restricted cases.

23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom