## Journal of Differential Geometry

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

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

Matthew D. Blair and Christopher D. Sogge

#### Abstract

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 $n \geq 3$.

#### Note

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

#### Article information

**Source**

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

**Dates**

Received: 4 November 2015

First available in Project Euclid: 23 May 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1527040871

**Digital Object Identifier**

doi:10.4310/jdg/1527040871

**Mathematical Reviews number (MathSciNet)**

MR3807318

**Zentralblatt MATH identifier**

06877018

**Subjects**

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]

**Keywords**

eigenfunctions Kakeya–Nikodym averages nodal sets

#### Citation

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. https://projecteuclid.org/euclid.jdg/1527040871