Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Second eigenvalue of a Jacobi operator of hypersurfaces with constant
  scalar curvature   Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with
constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere
$\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be
characterized as critical points for a variational problem of the integral
$\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal
upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$.
Moreover, when $r>1$, the bound is attained if and only if $M$ is totally
umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is
the Riemannian product
$\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq
n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.

9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom