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This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory…
We show that, in odd dimensions, any real valued, bounded potential of compact support has at least one scattering resonance. For dimensions three and higher this was previously known only for sufficiently smooth potentials. The proof is…
We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function…
On a compact three-dimensional Riemannian manifold with boundary, we prove the compactness of the full set of conformal metrics with positive constant scalar curvature and constant mean curvature on the boundary. This involves a blow-up…
We consider real isotropic geodesics on manifolds endowed with a pseudoconformal structure and their applications to the theory of lightlike hypersurfaces on such manifolds, the geometry of four-dimensional conformal structures of…
A uniqueness result in the inverse problem for an inhomogeneous hyperbolic system on a real vector bundle over a smooth compact manifold, based on energy measurements for improperly known sources, is established.
We consider the inverse scattering problem at fixed and sufficiently large energy for the nonrelativistic and relativistic Newton equation in $\R^n$, $n \ge 2$, with a smooth and short range electromagnetic field $(V,B)$. Using results of…
We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a $C^0$ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem…
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or…
We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the…
The spectral geometry of negatively curved manifolds has received more attention than its positive curvature counterpart. In this paper we will survey a variety of spectral geometry results that are known to hold in the context of…
We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already…
This paper makes a formal study of asymptotically hyperbolic Einstein metrics given, as conformal infinity, a conformal manifold with boundary. The space on which such an Einstein metric exists thus has a finite boundary in addition to the…
We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.
We study an inverse problem of determining a time-dependent potential appearing in the wave equation in conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that…
We conjecture that certain curvature invariants of compact hyperk\"ahler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an "experimental proof" in higher dimensions, and verify it for all known…
We consider inverse curvature flows in the $(n+1)$-dimensional Euclidean space, $n\geq 2,$ expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function $F$ with some concavity properties. We obtain asymptotical…
We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that,…
In this paper, we adapt the well-known \emph{local} uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schr{\"o}dinger equation to prove \emph{local} uniqueness results in the setting of inverse…
New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on…