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

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 $n\ge3$.