arXiv research

Maximal mean curvature limits surface area in higher dimensions.

In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This boun…

Study on maximal hypersurfaces in pp-wave spacetimes, proving non-existence and total geodesic properties.

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

Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.

Study of hypersurfaces in static spacetimes with curvature bounds.

Solves constant mean curvature properties in Reissner-Nordstrom 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 …

Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.

In this paper we prove that the H^k (k is odd and larger than 2) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total L^p integral of the mean curvature is finite for some p

Eigenfunction maps embed manifolds with a dimension bound.

We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the curvature of a the Lorentz boundary manifold.

Constructs solutions for gravitational instantons from minimal surfaces.

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter $N$-dimensional s…

The paper extends foliation by constant mean curvature surfaces to spacetimes with particles.

The paper solves splitting problems for specific spacetimes.

Study on total mean curvature and scalar curvature on 3-manifolds.

Study stable maximal hypersurfaces in various spacetimes.

Study of surfaces with zero mean curvature in a specific pseudo-Euclidean space.

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like to time-like at a given non-degenerate null curve. We consider this phenomenon a…

Rotationally symmetric hypersurfaces converge to cylinders under area-preserving flow.

Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…

3-manifolds with positive scalar curvature and bounded geometry are contractible.

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…

The paper studies how convex hypersurfaces evolve under a specific speed function of the mean curvature.

We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …

Let $V$ be a maximal globally hyperbolic flat $n+1$--dimensional space--time with compact Cauchy surface of hyperbolic type. We prove that $V$ is globally foliated by constant mean curvature hypersurfaces $M_τ$, with mean curvature $τ$ taking all values in $(-\infty, 0)$. For $n \geq 3$, define the rescaled volume of $…

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

The paper studies inverse mean curvature flow on hypersurfaces in space forms.

The paper shows how Scherk-type surfaces can be decomposed into helicoids.

Holomorphic discs converge to maximal surfaces under specific flows.

The study classifies surfaces of revolution in a specific Lorentzian product space.

New boundary condition for weak inverse mean curvature flow in bounded domains.

The main goal of this survey is to illustrate geometric applications of the Poincaré Lemma to constant mean curvature equations. In 1970, Calabi introduced the duality between minimal graphs in three dimensional Euclidean space and maximal graphs in three dimensional Lorentz space. We construct two extensions of Calabi…

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed…

New integrable systems for marginally trapped surfaces in 4D Lorentz-Minkowski space.

Proves that timelike maximal surfaces in (1+2) Minkowski space are always embedded.

We present a MAPLE program for the explicit computation of the curvature of calibrated ordinary webs in codimension one in any dimension (recall that all planar webs are calibrated and ordinary). The vanishing of this curvature means the maximality of the rank. We give some first examples.

Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.

We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singul…

Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.

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

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. Fo…

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

We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation extends simultaneously the Weierstrass representation for minimal surfaces in Euclidea…

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski 3-space R^3_1 have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been…