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Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial…
This paper presents results on the extent to which mean curvature data can be used to determine a surface in space or its shape. The emphasis is on Bonnet's problem: classify and study the surface immersions in $\R^3$ whose shape is not…
We survey some recent results on biconservative surfaces in $3$-dimensional space forms $N^3(c)$ with a special emphasis on the $c=0$ and $c=1$ cases. We study the local and global properties of such surfaces, from extrinsic and intrinsic…
We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of…
A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the…
In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change…
We prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second…
In this paper we classify complete surfaces of constant mean curvature whose Gaussian curvature does not change sign in a simply connected homogeneous manifold with a 4-dimensional isometry group.
We establish curvature inequalities and rigidity results for surfaces satisfying constant mean curvature type conditions in both Riemannian and Lorentzian geometry. In the Riemannian setting we study constant mean curvature (CMC) surfaces…
We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in…
We consider a compact, star-shaped, mean convex hypersurface $\Sigma^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well…
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss…
We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.
Some results about the geodesic boundary of minimal surfaces in $\mathbb{H}^2\times \mathbb{R}$ are generalized for surfaces of constant mean curvature surfaces $H$, with $0\le H\le 1/2$.
It is known that complex constant mean curvature ({\sc CMC} for short) immersions in $\mathbb C^3$ are natural complexifications of {\sc CMC}-immersions in $\mathbb R^3$. In this paper, conversely we consider {\it real form surfaces} of a…
Let $N$ be a Riemannian, Lorentzian or neutral $4$-dimensional space form with constant sectional curvature $L_0$. In this paper, noticing the linearly dependent condition, we obtain characterizations of space-like surfaces in $N$ with flat…
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 carry out the first main step towards the construction of new examples of complete embedded self-similar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of self-similar surfaces and…
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…
In this paper we classify compact minimal surfaces in $S^5$ with non-negative Gaussian curvature using the notion of a contact angle.