相关论文: Geometric Mean Curvature Lines on Surfaces Immerse…

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ where $({\mathbb R}^{m+p},\ol g)$ is the standard…

We exploit an arbitrary extrinsic time foliation of spacetime to solve the constraints in spherically symmetric general relativity. Among such foliations there is a one parameter family, linear and homogeneous in the extrinsic curvature,…

In this paper, we study factorable surfaces in a 3-dimensional isotropic space. We classify such surfaces with constant isotropic Gaussian (K) and mean curvature (H). We provide a non-existence result related with the surfaces satisfying…

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for…

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is…

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature.…

Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are…

We consider constant mean curvature surfaces of finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors in…

In this paper, we study the estimation of Gauss curvature for $K$-quasiconformal harmonic surface in ${\mathbb R}^3$ and present an accurate improvement of the previous result in [6, Theorem 5.2]. Let $X:M\rightarrow{\mathbb R}^3$ denote a…

We study the geometry of complete immersed surfaces in $\mathbb{R}^3$ with constant anisotropic mean curvature (CAMC). Assuming that the anisotropic functional is uniformly elliptic, we prove that: (1) planes and CAMC cylinders are the only…

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces…

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the…

A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact…

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a…

In this paper we prove that the only algebraic constant mean curvature (cmc) surfaces in R^3 of order less than four are the planes, the spheres and the cylinders. The method used heavily depends on the efficiency of algorithms to compute…

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff…

In this paper, we construct and classify minimal surfaces foliated by horizontal constant curvature curves in product manifolds $M \times \R$, where $M$ is the hyperbolic plane, the Euclidean plane or the two dimensional sphere. The main…

Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of ${\mathbb R}^4$. This includes the ellipsoidal…

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean…

We derive intrinsic curvature and radius estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature and apply these estimates to study the global geometry of complete surfaces embedded in $\mathbb{R}^3$ with…