Problems related to the concentration of eigenfunctions

We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for $L^1$-norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.Comment: 13 page

Concerning Nikodym-type sets in 3-dimensional curved spaces

We obtain estimates for maximal functions that arise when one studies Nikodym-type sets. We also formulate a curvature condition that allows favorable estimates for these maximal functions