Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variational characterizations of the total scalar curvature and
  eigenvalues of the Laplacian   For the dual operator $s_g'^*$ of the linearization $s_g'$ of the scalar
curvature function, it is well-known that if $\ker s_g'^*\neq 0$, then $s_g$ is
a non-negative constant. In particular, if the Ricci curvature is not flat,
then $ {s_g}/(n-1)$ is an eigenvalue of the Laplacian of the metric $g$. In
this work, some variational characterizations were performed for the space
$\ker s_g'^*$. To accomplish this task, we introduce a fourth-order elliptic
differential operator $\mathcal A$ and a related geometric invariant $\nu$. We
prove that $\nu$ vanishes if and only if $\ker s_g'^* \ne 0$, and if the first
eigenvalue of the Laplace operator is large compared to its scalar curvature,
then $\nu$ is positive and $\ker s_g'^*= 0$. Furthermore, we calculated the
lower bound on $\nu$ in the case of $\ker s_g'^* = 0$. We also show that if
there exists a function which is $\mathcal A$-superharmonic and the Ricci
curvature has a lower bound, then the first non-zero eigenvalue of the Laplace
operator has an upper bound.

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The categories of this paper are: cond-mat
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The categories of this paper are: eess
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con