Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness and Non-compactness for the Yamabe Problem on Manifolds With
  Boundary   We study the problem of conformal deformation of Riemannian structure to
constant scalar curvature with zero mean curvature on the boundary. We prove
compactness for the full set of solutions when the boundary is umbilic and the
dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under
these hypotheses, and we provide counter-examples to compactness when $n \geq
25$. Lastly, our methods point towards a vanishing theorem for the umbilicity
tensor, which is anticipated to be fundamental for a study of the nonumbilic
case.

16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an