Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal constants of $L^2$ inequalities for closed nearly umbilical
  hypersurfaces in space forms   Let $\Sigma$ be a smooth closed hypersurface with non-negative Ricci
curvature, isometrically immersed in a space form. It has been proved in
\cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on
$\Sigma$ which measure the stability of closed umbilical hypersurfaces or more
generally, closed hypersurfaces with traceless Newton transformation of the
second fundamental form. In this paper, we prove that the constants in these
inequalities are optimal.

23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
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The categories of this paper are: acc-phys
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics