Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Sharp Comparison Theorem for Compact Manifolds with Mean Convex
  Boundary   Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative
Ricci curvature and mean convex boundary $\partial M$. Assume that the mean
curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for
some positive constant $k$. In this paper, we prove that the distance function
$d$ to the boundary $\partial M$ is bounded from above by $\frac{1}{k}$ and the
upper bound is achieved if and only if $M$ is isometric to an $n$-dimensional
Euclidean ball of radius $\frac{1}{k}$.

10
The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
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The categories of this paper are: stat
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con