arXiv research

Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.

The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist …

We prove that the leaves of an inverse mean curvature flow provide a foliation of a future end of a cosmological spacetime $N$ under the necessary and sufficent assumptions that $N$ satisfies a future mean curvature barrier condition and a strong volume decay condition. Moreover, the flow parameter $t$ can be used to d…

We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing…

We consider spacetimes $N$ satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime $\hat N$ by switching the light cone and using reflection to define a new time function, such that the two…

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Further…

We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…

Study finds unique and non-existent constant mean curvature hypersurfaces in specific spacetimes.

We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called "time flat surfaces." S…

We solve the spacelike, spherically symmetric, constant mean curvature hypersurfaces in the maximally extended Reissner-Nordstrom spacetime with the charge smaller than the mass. Based on these results, we construct constant mean curvature foliations with fixed or varied mean curvature in each slice in this spacetime.

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

New method to find surfaces in null cones with constant curvature near black hole indicators.

Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.

Study existence of achronal hypersurfaces with prescribed mean curvature in 3D spacetimes.

Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.

Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.

Two rigidity results for surfaces in Schwarzschild spacetime.

This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

We prove that the leaves of the rescaled curvature flow considered in arXiv:math/0403485 [math.DG] converge to the graph of a constant function.

Paper proves uniqueness of specific spacetime surfaces in a lightcone.

Study of spacetimes in cosmology without symmetry assumptions.

The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward Bäcklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a generalized harmonic inverse mean curvature surface is constructed by a backward Bäck…

New CMC existence result for expanding cosmological spacetimes.

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining …

The paper estimates area and volume for spacetimes with integral mean curvature bounds.

The paper studies a flow of surfaces in spacetime with a focus on curvature evolution.

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the …

The paper establishes curvature inequalities and rigidity results for CMC and STCMC surfaces in Riemannian and Lorentzian geometry.

New results on non-existence and rigidity of spacelike submanifolds in spacetimes.

Constructs approximate mean curvature flows for general varifolds.

In this paper we study the r-stability of closed spacelike hypersurfaces with constant $r$-th mean curvature in conformally stationary spacetimes of constant sectional curvature. In this setting, we obtain a characterization of $r-$stability through the analysis of the first eigenvalue of an operator naturally attached…

The paper establishes curvature inequalities and rigidity results for surfaces in Riemannian and Lorentzian geometry.

We consider open globally hyperbolic spacetimes $N$ of dimension $n+1$, $n\ge 3$, which are spatially asymptotic to a Robertson-Walker spacetime or an open Friedmann universe with spatial curvature $\tildeκ= 0,-1$ and prove, under reasonable assumptions, that there exists a unique foliation by hypersurfaces of constant…

Let $M$ be a globally hyperbolic maximal compact $3$-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that $M$ admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularit…

Overview of marginally trapped surfaces in various spacetimes.

Proves existence of proper solutions for inverse mean curvature flow.

An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvatur…

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…

Proves higher regularity for anisotropic inverse mean curvature flow.

Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.

The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.

Study null energy condition impacts on special hypersurfaces in static spacetimes.

Rigidity results for hypersurfaces in warped spacetimes.

For $2+1$ spacetime dimensions, we derive sufficient conditions for the twisting function in a twisted product spacetime, such that there is a global foliation by spacelike CMC surfaces.

Inverse mean curvature flow converges to a disk in hyperbolic space.

We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003…