Related papers: Surfaces of prescribed linear Weingarten curvature…
We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in…
Wintgen proved in [P. Wintgen, Sur l'in\'egalit\'e de Chen-Willmore, C. R. Acad. Sci. Paris, 288 (1979), 993--995] that the Gauss curvature $K$ and the normal curvature $K^D$ of a surface in the Euclidean 4-space $E^4$ satisfy $$K+|K^D|\leq…
For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $\gamma$ projects this into the…
We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general…
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…
A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end…
We introduce a new class of surfaces in Euclidean $3$-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In…
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group.…
In Euclidean space we study surfaces with constant anisotropic mean curvature $\Lambda$ of the Dirichlet energy $\int_\Omega( |Du|^2+\Lambda u)$. We prove the existence of non-rotational surfaces with $\Lambda=0$ and foliated by a…
We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if…
We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to…
We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic)…
We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and…
We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…
An extension of the classic Enneper-Weierstrass representation for conformally parametrised surfaces in multi-dimensional spaces is presented. This is based on low dimensional CP^1 and CP^2 sigma models which allow the study of the constant…
We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S^3, H^3 and partial results…
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 stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of…
A novel class of integrable surfaces is recorded. This class of O surfaces is shown to include and generalize classical surfaces such as isothermic, constant mean curvature, minimal, `linear' Weingarten, Guichard and Petot surfaces and…
In this paper we study constant angle surfaces in Euclidean 3-space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the…