Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solving the Yamabe Problem on Closed Manifolds by Iteration Schemes and
  Perturbation Methods   We introduces a double iterative scheme and a perturbation method to solve
the Yamabe equation $ \Box_{g}u : = - \frac{4(n - 1)}{n - 2}\Delta_{g} u +
S_{g} u = \lambda u^{\frac{n + 2}{n - 2}} $ on closed manifolds $ (M, g) $ with
$ \dim M \geqslant 3 $. Thus $g$ admits a conformal change to a constant scalar
curvature metric. In contrast to the traditional method, the Yamabe problem is
completely solved in five cases classified by the sign of the scalar curvature
$ S_{g} $ and the sign of the first eigenvalue of conformal Laplacian. In
contrast to the traditional proof, we focus on local analysis; both Weyl tensor
and positive mass theorem are not required, and the proof is essentially
dimension independent. To treat the most difficult case $ \lambda > 0 $, we
construct the lower and upper solutions to solve a perturbed Yamabe equation $
\Box_{g} u_{\beta} + \beta u_{\beta} = \left( \lambda_{\beta} - \kappa \right)
u_{\beta}^{\frac{n + 2}{n - 2}} $ with $ \kappa > 0 $, $ \beta < 0 $ and some
specific choices of $ \lambda_{\beta}'s $, then take a limit by letting $ \beta
\rightarrow 0 $.

12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci