## 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

### Keywords:

eigenfunctions; Kakeya-Nikodym averages; nodal sets
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