We define the forward and backward radiation fields on an asymptotically hyperbolic manifold and show that they give unitary translation representations of the wave group and as such can be used to define a scattering matrix. We show that this scattering matrix is equivalent to the one defined by stationary methods. Furthermore, we prove a support theorem for the radiation fields which generalizes to this setting well-known results of Helgason [23] and Lax and Phillips [35] for the horocyclic Radon transform. As an application, we use the boundary control method of Belishev [4] to show that an asymptotically hyperbolic manifold is determined up to invariants by the scattering matrix at all energies.

There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.

We prove sharp $L^p$ Carleman estimates and the corresponding unique continuation results for second-order real principal-type differential equations $P(x,D)u+V(x)u=0$ with critical potential $V\in L_{\mathrm{loc}}^{n/2}$ (where $n\ge 3$ is the dimension) across a noncharacteristic hypersurface under a pseudoconvexity assumption. Similarly, we prove unique continuation results for differential equations with potential in the Calderón uniqueness theorem's context under a curvature condition.

We also investigate ($L^p-L^{p\text{'}}$)-estimates for non-self-adjoint pseudodifferential operators under a curvature condition on the characteristic set and develop the natural applications to local solvability for the corresponding operators with potential.

We look at complete, locally conformally flat (lcf) metrics defined on a domain $\Omega \subset S^n$. There is a lot of information about the singular set $\partial \Omega$ contained in the positivity of ${\sigma }_k$, and, in particular, we obtain a bound for the Hausdorff dimension of $\partial \Omega$ in relation to Schoen and Yau's work [18] for the scalar curvature. On the other hand, since some locally conformally flat manifolds can be embedded into $S^n$, this dimension bound implies several topological corollaries.

In this article we study the local geometry at a prime $p$ of PEL-type Shimura varieties for which there is a hyperspecial level subgroup. We consider the Newton polygon stratification of the special fiber at $p$ of Shimura varieties and show that each Newton polygon stratum can be described in terms of the products of the reduced fibers of the corresponding PEL-type Rapoport-Zink spaces with certain smooth varieties (which we call Igusa varieties) and of the action on them of a $p$-adic group that depends on the stratum. We then extend our construction to characteristic zero and, in the case of bad reduction at $p$, use it to compare the vanishing cycle sheaves of the Shimura varieties to those of the Rapoport-Zink spaces. As a result of this analysis, in the case of proper Shimura varieties we obtain a description of the $l$-adic cohomology of the Shimura varieties in terms of the $l$-adic cohomology with compact supports of the Igusa varieties and of the Rapoport-Zink spaces for any prime $l\ne p$.