Related papers: Prescribing Mean Curvature: Existence and Uniquene…
A generic surface in Euclidean 3-space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known…
Global isothermic immersions are defined and studied with the aid of a connection between quadratic differentials and immersions. The applications are two problems stemming from the fundamental question: how much data is needed to identify…
For a surface immersed in a three-dimensional space endowed with a norm instead of an inner product, one can define analogous concepts of curvature and metric. With these concepts in mind, various questions immediately appear. The aim of…
In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simply-connected ones. We consider the isometric deformability question for an immersion x: M \to R^3…
Here are studied qualitative properties of the families of curves --foliations-- on a surface immersed in ${\mathbb R}^4$, along which it bends extremally in the direction of the mean normal curvature vector. Typical singularities and…
We study the Bonnet problem in Bianchi--Cartan--Vr\u{a}nceanu spaces and in $\mathrm{Sol}_3$. Our main contribution is to establish the uniqueness of Bonnet mates, which leads us to address the problem of determining when an isometric…
The aim of this paper is to extend classic results of the theory of CMC surfaces in the product spaces to the class of immersed surfaces in $\mathbb{M}^2(\kappa)\times\mathbb{R}$ whose mean curvature is given as a $C^1$ function depending…
Associated to oriented surfaces immersed in R^3 here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k_1, k_2…
We present a method giving a spinorial characterization of an immersion in a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory in that spaces. We…
The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(\kappa)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends…
Motivated by the large ammount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in $\mathbb{R}^3$…
In this paper we solve the Bj\"orling problem for the class of immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with…
In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…
We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the…
We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms $\h^3$ and $\s^3$ to be contained in a totally umbilical surface. In case that the curve has constant torsion, we obtain the…
In this paper we study Moebius applicable surfaces, i.e., conformally immersed surfaces in Moebius 3-space which admit deformations preserving the Moebius metric. We show new characterizations of Willmore surfaces, Bonnet surfaces and…
A Killing submersion is a Riemannian submersion from a 3-manifold to a surface, both connected and orientable, whose fibres are the integral curves of a Killing vector field, not necessarily unitary. The first part of this paper deals with…
We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When…
We show short time existence and uniqueness of $\C^{1,1}$ solutions to the mean curvature flow with obstacles, when the obstacles are of class $\C^{1,1}$. If the initial interface is a periodic graph we show long time existence of the…
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean…