Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Large conformal metrics with prescribed scalar curvature   Let $(M,g)$ be an $n-$dimensional compact Riemannian manifold. Let $h$ be a
smooth function on $M$ and assume that it has a critical point $\xi\in M$ such
that $h(\xi)=0$ and which satisfies a suitable flatness assumption. We are
interested in finding conformal metrics $g_\lambda=u_\lambda^\frac4{n-2}g$,
with $u>0$, whose scalar curvature is the prescribed function
$h_\lambda=\lambda^2+h$, where $\lambda$ is a small parameter.
  In the positive case, i.e. when the scalar curvature $R_g$ is strictly
positive, we find a family of bubbling metrics $g_\lambda$, where $u_\lambda$
blows-up at the point $\xi$ and approaches zero far from $\xi$ as $\lambda$
goes to zero.
  In the general case, if in addition we assume that there exists a
non-degenerate conformal metric $g_0=u_0^\frac4{n-2}g$, with $u_0>0$, whose
scalar curvature is equal to $h$, then there exists a bounded family of
conformal metrics $g_{0,\lambda}=u_{0,\lambda}^\frac4{n-2}g$, with
$u_{0,\lambda}>0$, which satisfies $u_{0,\lambda}\to u_0$ uniformly as
$\lambda\to 0$. Here, we build a second family of bubbling metrics $g_\lambda$,
where $u_\lambda$ blows-up at the point $\xi$ and approaches $u_0$ far from
$\xi$ as $\lambda$ goes to zero. In particular, this shows that this problem
admits more than one solution.

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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: supr-con
16
The categories of this paper are: stat
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn