Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Poincar\'e inequality on minimal graphs over manifolds and applications   Let $B_2(p)$ be an $n$-dimensional smooth geodesic ball with Ricci curvature
$\geq-(n-1)\kappa^2$ for some $\kappa\geq0$. We establish the Sobolev
inequality and the uniform Neumann-Poincar\'e inequality on each minimal graph
over $B_1(p)$ by combining Cheeger-Colding theory and the current theory from
geometric measure theory, where the constants in the inequalities only depends
on $n$, $\kappa$, the lower bound of the volume of $B_1(p)$. As applications,
we derive gradient estimates and a Liouville theorem for a minimal graph $M$
over a smooth complete noncompact manifold $\Sigma$ of nonnegative Ricci
curvature and Euclidean volume growth. Furthermore, we can show that any
tangent cone of $\Sigma$ at infinity splits off a line isometrically provided
the graphic function of $M$ admits linear growth.

17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc