Journal of Differential Geometry

Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions

Matthew D. Blair and Christopher D. Sogge

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We use Toponogov’s triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya–Nikodym norms introduced in [22] for manifolds of nonpositive sectional curvature. Using these and results from our paper [4] we are able to obtain log-improvements of $L^p (M)$ estimates for such manifolds when $2 \lt p \lt \frac{2(n+1)}{n-1}$. These in turn imply $(\log \lambda)^{\sigma_n} , \sigma_n \approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch [28], and using a result from Hezari and the second author [18], under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi [12] by a factor of $(\log \lambda)^{\mu}$ for any $\mu \lt \frac{2(n+1)^2}{n-1}$, if $n \geq 3$.


The authors were supported in part by the NSF grants DMS-1301717 and DMS-1361476, respectively.

Article information

J. Differential Geom., Volume 109, Number 2 (2018), 189-221.

Received: 4 November 2015
First available in Project Euclid: 23 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
Secondary: 35A99: None of the above, but in this section 42B37: Harmonic analysis and PDE [See also 35-XX]

eigenfunctions Kakeya–Nikodym averages nodal sets


Blair, Matthew D.; Sogge, Christopher D. Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions. J. Differential Geom. 109 (2018), no. 2, 189--221. doi:10.4310/jdg/1527040871.

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