On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature

By Christopher D. Sogge, Steve Zelditch. 2011

55 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d2db191ddccbea41f50d6313f0d6069b86c25b4

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Citations in the corpus

Improvement of eigenfunction estimates on manifolds of nonpositive curvature (2012, background): “||L2(γ) where γis a geodesic. Sogge shows that to improve the Lp(M) estimates for 2 <p<6 it is necessary and suﬃcient to improve the L2 restriction estimates. In related work Sogge and Zelditch [11] and Chen and Sogge [6] study improvements to geodesic restriction theorems for low pin the case”
A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions (2012, methodology): “ for the size of nodal sets of eigenfunctions (see [11], [33] and [16]). In the main, in establishing (1.5), we follow the template provided the proof of the earlier corresponding weaker estimates in [35] where the L4(γ)-norms are replaced by L2(γ) ones, except instead of using microlocal analysis t”