Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Prescribing curvatures on three dimensional Riemannian manifolds with
  boundaries   Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary
$\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and
$\partial M$, respectively, it is natural to ask whether there exist metrics
conformal to $g$ so that under these new metrics, $K$ is the scalar curvature
and $c$ is the boundary mean curvature. All such metrics can be described by a
prescribing curvature equation with a boundary condition. With suitable
assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the
equation can only blow up at finite points over each compact subset of $\bar
M$, some of them may appear on $\partial M$. We describe the asymptotic
behavior of the blowup solutions around each blowup point and derive an energy
estimate as a consequence.

13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg