Let $(M,g)$ be a smooth, compact Riemannian manifold, and let $\{{\varphi }_h\}$ be an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2{\Delta }_g{\varphi }_h={\varphi }_h$. Given a smooth submanifold $H\subset M$ of codimension $k\ge 1$, we find conditions on the pair $(\{{\varphi }_h\},H)$ for which $$\left|{\int }_H{\varphi }_{h}d{\sigma }_H\right|=o\left(h^{\frac{1-k}{2}}\right),h\to 0^{+}.$$ One such condition is that the set of conormal directions to $H$ that are recurrent has measure $0$. In particular, we show that the upper bound holds for any $H$ if $(M,g)$ is a surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.

We derive extrinsic curvature estimates for compact disks embedded in ${\mathbb{R}}^3$ with nonzero constant mean curvature.

Let $\pi$ be a discrete series representation of a real reductive Lie group $G\text{'}$, and let $G$ be a reductive subgroup of $G\text{'}$. In this paper, we give a geometric expression of the $G$-multiplicities in $\pi {|}_G$ when the representation $\pi$ is $G$-admissible.

This paper studies the sliced nearby cycle functor and its commutation with duality. Over a Henselian discrete valuation ring, we show that this commutation holds, confirming a prediction of Deligne. As an application we give a new proof of Beilinson’s theorem that the vanishing cycle functor commutes with duality up to twist. Over an excellent base scheme, we show that the sliced nearby cycle functor commutes with duality up to modification of the base. We deduce that duality preserves universal local acyclicity over an excellent regular base. We also present Gabber’s theorem that local acyclicity implies universal local acyclicity over a Noetherian base.