Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On uniqueness and nonuniqueness of ancient ovals   In this paper, we prove that any nontrivial $\mathrm{SO}(k )\times
\mathrm{SO}(n+1-k)$-symmetric ancient compact noncollapsed solution of the mean
curvature flow agrees up to scaling and rigid motion with the
$\mathrm{O}(k)\times \mathrm{O}(n+1-k)$-symmetric ancient ovals constructed by
Hershkovits and the second author. This confirms a conjecture by
Angenent-Daskalopoulos-Sesum. On the other hand, for every $k\geq 2$ we also
construct a $(k-1)$-parameter family of uniformly $(k+1)$-convex ancient ovals
that are only $\mathbb{Z}^{k}_{2}\times \mathrm{O}(n+1-k)$-symmetric. This
gives counterexamples to a conjecture of Daskalopoulos.

1
The categories of this paper are: cs
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn