Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ancient low entropy flows, mean convex neighborhoods, and uniqueness   In this article, we prove the mean convex neighborhood conjecture for the
mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a
spherical or cylindrical singularity at a space-time point $X=(x,t)$, then
there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is
mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The
major difficulty is to promote the infinitesimal information about the
singularity to a conclusion of macroscopic size. In fact, we prove a more
general classification result for all ancient low entropy flows that arise as
potential limit flows near $X$. Namely, we prove that any ancient,
unit-regular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at
most $\sqrt{2\pi/e}+\delta$ is either a flat plane, a round shrinking sphere, a
round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an
application, we prove the uniqueness conjecture for mean curvature flow through
spherical or cylindrical singularities. In particular, assuming Ilmanen's
multiplicity one conjecture, we conclude that for embedded two-spheres the mean
curvature flow through singularities is well-posed.

23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn