# Averages of eigenfunctions over hypersurfaces

## TL;DR

This paper investigates how eigenfunctions of the Laplacian behave when restricted to hypersurfaces on a Riemannian manifold, showing that under certain conditions, their integrals tend to zero as the eigenvalues grow.

## Contribution

It establishes a new result relating the defect measure's concentration properties to the vanishing of eigenfunction restrictions on hypersurfaces.

## Key findings

- Eigenfunction restrictions to hypersurfaces tend to zero under non-concentration conditions.
- Normal derivatives of eigenfunctions also tend to zero on hypersurfaces.
- The result links measure concentration to eigenfunction behavior in the high-frequency limit.

## Abstract

Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ \phi_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $\mu$. Let $H$ be a smooth hypersurface. Our main result says that when $\mu$ is $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.