Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions

TL;DR

This paper leverages Toponogov's triangle comparison theorem and propagation of singularities to achieve logarithmic improvements in eigenfunction estimates and nodal set bounds on manifolds with nonpositive curvature.

Contribution

It introduces new logarithmic enhancements of eigenfunction and eigenvalue estimates using geometric comparison theorems and advanced analysis techniques.

Findings

Logarithmic improvements of Kakeya-Nikodym norms.

Enhanced $L^p$ eigenfunction estimates with logarithmic factors.

Stronger lower bounds on eigenfunction $L^1$ norms and nodal set sizes.

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 \cite{SKN} for manifolds of nonpositive sectional curvature. Using these and results from our paper \cite{BS15} we are able to obtain log-improvements of $L^p(M)$ estimates for such manifolds when $2<p<\tfrac{2(n+1)}{n-1}$. 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~\cite{SZ11}, and using a result from Hezari and the second author~\cite{HS}, under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi~\cite{CM} by a factor of $(\log \lambda)^{\mu}$ for any $\mu<\tfrac{2(n+1)^2}{n-1}$, if $n\ge3$.