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June 2018 Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions
Matthew D. Blair, Christopher D. Sogge
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J. Differential Geom. 109(2): 189-221 (June 2018). DOI: 10.4310/jdg/1527040871


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 $n \geq 3$.

Funding Statement

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


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Matthew D. Blair. Christopher D. Sogge. "Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions." J. Differential Geom. 109 (2) 189 - 221, June 2018.


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

zbMATH: 06877018
MathSciNet: MR3807318
Digital Object Identifier: 10.4310/jdg/1527040871

Primary: 58J51
Secondary: 35A99 , 42B37

Keywords: Eigenfunctions , Kakeya–Nikodym averages , nodal sets

Rights: Copyright © 2018 Lehigh University


Vol.109 • No. 2 • June 2018
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