Related papers: On Lorentzian surfaces in $\mathbb{R}^{2,2}$
We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant…
We study the motion of smooth, strictly convex bodies in $\mathbb{R}^n$ expanding in the direction of their normal vector field with speed depending on Gauss curvature and support function.
We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the…
We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently:…
In this paper, we study third order nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness of connection 1-forms, we present a classification of equations with the type $u_t - u_{xxt} =…
A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give…
In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and…
In this work, we seek characterizations of global hyperbolicity in smooth Lorentzian manifolds that do not rely on the manifold topology and that are inspired by metric geometry. In particular, strong causality is not assumed, so part of…
In this paper, we study the Lorentzian minimal surfaces in the Minkowski space-time with finite type Gauss map. First, we obtain the classification of this type of surfaces with pointwise 1-type Gauss map. Then, we proved that there are no…
We address the asymptotic behavior of the $\alpha$-Gauss curvature flow, for $\alpha >1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow…
In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We…
In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under…
We study the geometry of Nieto's quintic threefold (Barth & Nieto, J. Alg. Geom. 3, 1994) and the Kummer and abelian surfaces that correspond to special loci.
In this work, we study some classes of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}^4_t$ with profile curves lying in 2-dimensional planes. First, we determine all such surfaces in the Minkowski 4-space $\mathbb{E}^4_1$…
We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three…
We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general…
We consider an asymptotically flat Lorentzian manifold of dimension (1,3). An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality…
We consider a surface embedded in the Euclidean 3-space and fix a tangential vector $v$ at a given point $p$ on the surface. In this paper, we first review a history of the formula obtained by Mannheim, d'Ocagne and Koenderink, which…
We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional surfaces. We consider one and two parameter general unfoldings…
In this text we expound recent results by Idrisse Khemar on the construction of various geometric completely integrable systems generalizing the structure of Hamiltonian stationary Lagrangian surfaces (HSLS) discovered by F. H\'elein and P.…