Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On eigenfunction restriction estimates and $L^4$-bounds for compact
  surfaces with nonpositive curvature   Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold
with nonpostive curvature, then we shall give improved estimates for the
$L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics,
compared to the general results of Burq, G\'erard and Tzvetkov \cite{burq}. By
earlier results of Bourgain \cite{bourgainef} and the first author
\cite{Sokakeya}, they are equivalent to improvements of the general
$L^p$-estimates in \cite{soggeest} for $n=2$ and $2<p<6$. The proof uses the
fact that the exponential map from any point in $x_0\in M$ is a universal
covering map from $\Rt \simeq T_{x_0}M$ to $M$ (the Cartan-Hadamard- von
Mangolt theorem), which allows us to lift the necessary calculations up to the
universal cover $(\Rt, \tilde g)$ where $\tilde g$ is the pullback of $g$ via
the exponential map. We then prove the main estimates by using the Hadamard
parametrix for the wave equation on $(\Rt, \tilde g)$ and the fact that the
classical comparison theorem of G\"unther \cite{Gu} for the volume element in
spaces of nonpositive curvature gives us desirable bounds for the principal
coefficient of the Hadamard parametrix, allowing us to prove our main result.

18
The categories of this paper are: econ
19
The categories of this paper are: eess
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph