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

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 $n\ge3$.