arXiv research

The paper proves rigidity results for compact initial data sets.

Proves rigidity for specific initial data sets under the dominant energy condition.

We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No…

Proves density and mass theorems for specific initial data sets.

We prove in this paper that, under suitable coinditions on an initial data set, we can obtain Area and Curvature Estimates for simple marginally outer trapped surfaces (or MOTS). Using this estimates, we derive a Compactness Theorem for MOTS. Moreover, the Compactness Theorem will allow us to adapt the recent Degree Th…

We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conf…

The study extends conserved quantities theory to non-compact boundary initial data sets.

The paper proves positive energy-momentum theorems for charged AdS initial data sets.

Paper proves new inequalities for Einstein-Maxwell data sets.

We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desi…

Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.

We construct large families of initial data sets for the vacuum Einstein equations with positive cosmological constant which contain exactly Delaunay ends; these are non-trivial initial data sets which coincide with those for the Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the purely Rie…

Study finds rigid properties of boundary-free hypersurfaces in specific data sets.

Constructs initial data for Einstein equations and estimates Bartnik mass outside time-symmetry.

Proves properties of free boundary stable MOTS in spacetimes.

In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positi…

In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry a non trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces have constant mean curvature and are isometric to one of the following manifolds: (i) Sn the standard spher…

Proves stability of Minkowski space for specific initial data.

We introduce a natural generalization of marginally outer trapped surfaces, called immersed marginally outer trapped surfaces, and prove that three dimensional asymptotically flat initial data sets either contain such surfaces or are diffeomorphic to R^3. We establish a generalization of the Penrose singularity theorem…

In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in the plane. If $K$ is locally-connected, connected and compact, then the level set…

We establish both local and global well-posedness for the heat flow of polyharmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data with small BMO norms.

We investigate the well-posedness of (i) the heat flow of harmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow $(u,d)$ of nematic liquid crystals on $\mathbb R^n$ for initial data in $BMO^{-1}\times BMO$.

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split s…

Paper proves rigidity of initial data sets with boundary and capillary MOTS.

When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …

Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2 symmetry, establish the global well-posedness of the initial value problem for Einstein's field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity ass…

We develop a gluing construction which adds scaled and truncated asymptotically Euclidean solutions of the Einstein constraint equations to compact solutions with potentially non-trivial cosmological constants. The result is a one-parameter family of initial data which has ordinary and scaled "point-particle" limits an…

In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in irreducible symmetric spaces of compact type occur as principal orbits of Herman…

Constructs constant spacetime mean curvature surfaces for hyperboloidal initial data sets.

For convex co-compact hyperbolic quotients $X=Γ\backslash\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the Hausdorff dimension $δ$ of the limit set is less than $n/2$, then $u(t) = C_δ(f) e^{(δ-\nd…

Estimates bandwidth for CMC initial data sets.

Smooth dec initial data sets may not extend to smooth spacetimes.

We solve Bartnik's stationary extension problem near Schwarzschild spheres.

In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of \textsl{possibly non-compact} marginally outer-trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable margi…

Proves critical points of ADM mass correspond to specific initial data sets.

We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in any case, and generically, the gluing procedure can be l…

The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.

New PDE systems generalize Hawking mass monotonicity.

Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets

Initial data with zero mass must be in pp-wave spacetimes.

Proves principles and estimates for initial data sets in Einstein equations.

Study proposes initial data sets for solving gravitational equations, proving energy estimates.

Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.

Solves generalized Kähler Calabi-Yau problem on compact manifolds.

Rigidity results for initial data sets related to the positive mass theorem.

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 energy condition, the future Cauchy development of the data is a black hole spa…

We address the asymptotic behavior of the $α$-Gauss curvature flow, for $α>1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determ…

Constructs initial data for Einstein vacuum equations involving multiple localized gravitational sources.