Marc Soret

Mixed gravitational field equations on globally hyperbolic spacetimes

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting T (M) = D ⊕ R∇T of M, as exhibited by Bernal and S´ anchez (2005 Commun. Math. Phys. 257 43–50). We give sufficient geometric conditions (e.g. T is isoparametric and D is totally umbilical) for the existence of exact solutions −β dT ⊗ dT + g to mixed field equations in free space. We linearize and solve the mixed field equations RicD(g)μν − ρD(g) gμν = 0 for empty space, where ρD(g) is the mixed scalar curvature of foliated spacetime (M, D) (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If g� = g0 + �γ is a solution to the linearized field equations, then each leaf of D is totally geodesic in (R 4 \ R, g� ) to order O(�) . We derive the equations of motion of a material particle in the gravitational field gμν governed by the mixed field equations RicD(g)μν − ρD(g )ω μων − � gμν = 2 πκ c −2 Tμν − 1 Tg μν . In the weak field (� � 1) and low velocity (� v� /c � 1) limit, the motion equations are d 2 r/dt 2 =∇ φ + F, where φ = (�/ 2)c 2 γ00.

Maximum principle at infinity for complete minimal surfaces in flat 3-manifolds

The main result of this paper is the following maximum principle at infinity:Theorem.Let M1and M2be two disjoint properly embedded complete minimal surfaces with nonempty boundaries, that are stable in a complete flat 3-manifold. Then dist(M1,M2)=min(dist(∂M1,M2), dist(∂M2,M1)).In case one boundary is empty, e.g. M1,then dist(M1,M2)=dist(∂M2,M1).If both boundaries are empty, then M1and M2are flat.

Lissajous and Fourier knots

We prove that any knot of

b-Completion of pseudo-Hermitian manifolds

\mathbb{R}^3

SINGULARITY KNOTS OF MINIMAL SURFACES IN ℝ4

is isotopic to a Fourier knot of type

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

(1,1,2)

New minimal surfaces in S^3 desingularizing the Clifford tori

obtained by deformation of a Lissajous knot.

The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold

We study the interrelation among pseudo-Hermitian and Lorentzian geometry as prompted by the existence of the Fefferman metric. Specifically for any nondegenerate Cauchy–Riemann manifold M we build its b-boundary ˙ M .T his arises as a S 1 quotient of the b-boundary of the (total space of the canonical circle bundle over M endowed with the) Fefferman metric. Points of ˙ M are shown to be endpoints of b-incomplete curves. A class of inextensible integral curves of the Reeb vector on a pseudo-Einstein manifold is shown to have an endpoint on the b-boundary provided that the horizontal gradient of the pseudoHermitian scalar curvature satisfies an appropriate boundedness condition. Dedicated to the memory of Stere Ianus ¸ 4

Stably embedded minimal hypersurfaces

We study knots in 𝕊3 obtained by the intersection of a minimal surface in ℝ4 with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.