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A mathematical model of the evolution of spherical perturbations in a cosmological ideal scalar-charged fluid with scalar Higgs interaction is constructed. A closed mathematical model of linear spherical perturbations in a cosmological…
We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of…
A D-dimensional gravitational model with a Gauss-Bonnet term and the cosmological term Lambda is considered. By assuming diagonal cosmological metrics, we find, for certain fine-tuned Lambda, a class of solutions with exponential time…
We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal…
We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this…
The aim of this work is to deal with a discontinuous Hamilton-Jacobi equation in the whole euclidian N-dimensional space, associated to a possibly unbounded optimal control problem. Here, the discontinuities are located on a hyperplane and…
Recent works by the second author and Maxwell et al. have shown that the Einstein-scalar field conformal constraint equations are highly complex and generally intractable, even in the vacuum case. In this article, to gain a clearer…
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…
We develop a non-linear distributional renormalisation algebra for Gaussian Quantum Foam, built from sequences of scaled Gaussians on spacelike hypersurfaces of homotopic, globally hyperbolic spacetimes and their distributional limits. The…
We prove that if $\Sigma$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation…
We give a complete classification of isoparametric hypersurfaces in a product space $M^2_{\kappa_1}\times M^2_{\kappa_2}$ of $2$-dimensional space forms for $\kappa_i\in \{-1,0,1\}$ with $\kappa_1\neq \kappa_2$. In fact we prove that any…
We present a novel supersymmetric solution to a nonlinear sigma model coupled to supergravity. The solution represents a static, supersymmetric, codimension-two object, which is different to the familiar cosmic strings. In particular, we…
In the context of sub-Riemannian Heisenberg groups Hn, n \geq 1, we shall study Isoperimetric Profiles, which are closed compact hypersurfaces having constant horizontal mean curvature, very similar to ellipsoids. Our main goal is to study…
It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact…
We classify weakly complete constant Gaussian curvature $-1<K<0$ surfaces in the hyperbolic three-space in terms of holomorphic quadratic differentials. For this purpose, we first establish a loop group method for constant Gaussian…
We prove that any weakly acausal curve $\Gamma$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$.…
In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs…
We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.
We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature…
We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the…