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arXiv · 1510.07726

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

Matthew D. Blair·Christopher D. Sogge
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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 $(\logλ)^{σ_n}$, $σ_n\approx n$, 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 $(\log λ)^μ$ for any $μ<\tfrac{2(n+1)^2}{n-1}$, if $n\ge3$.

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Matthew D. Blair, Christopher D. Sogge. 2015-10-28. Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions. https://arxiv.org/abs/1510.07726

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