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Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions. (English) Zbl 1394.58016

Let \((M,g)\) be a compact Riemannian manifold without boundary of dimension \(m\geq2\) with injectivity radius at least 10. Let \(e_\lambda\) be an eigenfunction of the scalar Laplacian corresponding to the eigenvalue \(\lambda^2\) with \(\|e_\lambda\|_{L^2}=1\). Let \(\Pi\) be the space of all geodesic segments of length 1. Let \(T_\epsilon(\gamma)\) be the geodesic tube of width \(\epsilon\) about a geodesic \(\gamma\) in \(\Pi\).
Theorem: If \((M,g)\) has nonpositive sectional curvature, then there exists \(\kappa\) and \(\lambda_0\) so that if \(\lambda\geq\lambda_0\), then \[ \left\{\sup_{\gamma\in\Pi}\int_{T_{\lambda^{-\frac12}}(\gamma)}|e_\lambda|^2\right\}\leq\kappa c(\lambda) \] where \[ c(\lambda)=\begin{cases} (\log\lambda)^{-\frac12} \text{ if } m=2\\ (\log\lambda)^{-1}\log\log\lambda \text{ if }m=3\\ (\log\lambda)^{-1}\text{ if }m\geq4\end{cases}. \] Furthermore, if \(m=2\), then \(\sup_{\gamma\in\Pi}\int_\gamma|e_\lambda|^2\leq C\lambda^{\frac12}c(\lambda)\).
This estimate is a logarithmic improvement over previous estimates. Similar estimates are obtained in \(L^p\) for \(p\) suitably chosen.

MSC:

58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
35A99 General topics in partial differential equations
42B37 Harmonic analysis and PDEs
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