Related papers: Some rigidity results for charged initial data set…
For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $\gamma$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times…
Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that…
The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into…
Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null…
We construct the marginal loosely trapped surface (marginal LTS) for the Kerr spacetime with a small Kerr parameter perturbatively, where the LTS condition is saturated. An LTS is a surface that specifies the strong gravity region, which is…
We study the evolution of horizons of black holes in the $1+1+2$ covariant setting and investigate various properties intrinsic to the geometry of the foliation surfaces of these horizons. This is done by interpreting formulations of…
The construction of initial-data sets representing binary black-hole configurations in quasi-circular orbits is studied in the context of the conformal-imaging formalism. An effective-potential approach for locating quasi-circular orbits is…
We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of \(T^{\rtimes}\)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary…
Boundary charges in gauge theories (like the ADM mass in general relativity) can be understood as integrals of linear conserved n-2 forms of the free theory obtained by linearization around the background. These forms are associated…
In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for a 3-dimensional initial data set $(M,g,K)$ with mean curvature $\tau\le0$ in a 4-dimensional spacetime $(\overline M,\overline g)$ when (under suitable…
As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4\pi/c$,…
We prove a harmonic asymptotics density theorem for asymptotically flat initial data sets with compact boundary that satisfy the dominant energy condition. We use this to settle the spacetime positive mass theorem, with rigidity, for…
In this paper, we perform a detailed investigation on the various geometrical properties of trapped surfaces and the boundaries of trapped region in general relativity. This treatment extends earlier work on LRS II spacetimes to a general 4…
We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem…
We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the…
We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature…
Marginally outer trapped surfaces are widely considered as the best quasi-local replacements for event horizons of black holes in General Relativity. However, this equivalence is far from being proved, even in stationary and static…
We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal marginally trapped…
In this paper a lower bound for the ADM mass is given in terms of the angular momenta and charges of black holes present in axisymmetric initial data sets for the Einstein-Maxwell equations. This generalizes the mass-angular momentum-charge…
We construct initial data suitable for the Kerr stability conjecture, that is, solutions to the constraint equations on a spacelike hypersurface with boundary entering the black hole horizon that are arbitrarily decaying perturbations of a…