We solve the Jang equation with respect to asymptotically hyperbolic ``hyperboloidal'' initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.
We give a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization. The construction can be regarded as a "lattice approximation of the correspondence between differential operato...
We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the Atiyah-Patodi-Singer theorem that relates the index to the bulk integral of Pontryagin density and $\eta$-invariants of auxil...
We propose a general procedure to construct noncommutative deformations of an embedded submanifold $M$ of $\mathbb{R}^n$ determined by a set of (smooth or polynomial) equations $f^a(x)=0$. We use the framework of Drinfel'd twist deformation of differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006), 1883]; the commutative pointw...
We consider complex manifolds with holomorphically trivial canonical bundle endowed with a balanced metric. In the compact case, such manifolds are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Strominger system. Let $M$ be a compact simply connected balanced cohomogeneity one 6-manifold end...
We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D. Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at least one Killing vector. We rederive the results of Przanowski and collaborators, that these metrics can all b...
In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the $\sigma_{2}$ curvature has lower bound $\sigma_{2}\geq\frac{3}{2}$, we prove a Penrose type inequality relating the mass and contributions from singularities. We a...
Lecture notes written for a one-semester course in mathematical relativity aimed at mathematics and physics students. Not meant as an introduction to general relativity, but rather as a complementary, more advanced text.
Berwald spacetimes are Finsler spacetimes that are closest to pseudo-Riemannian spacetime geometry. Applying the cosmological principle, we find the most general spatially homogeneous and isotropic Berwald spacetime geometries. They are defined by a Finsler Lagrangian built from a free function on spacetime and a zero-homogeneous function on the ta...
We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent ...
