Related papers: Curvature estimates for surfaces with bounded mean…

In this paper are studied the nets of principal curvature lines on surfaces embedded in Euclidean $3-$space near their end points, at which the surfaces tend to infinity. This is a natural complement and extension to smooth surfaces of the…

We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures…

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve, in [11]. We obtain the Frenet apparatus of these…

We introduce two invariants called the secondary cuspidal curvature and the bias on $5/2$-cuspidal edges, and investigate their basic properties. While the secondary cuspidal curvature is an analog of the cuspidal curvature of (ordinary)…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

Let $\Omega\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_{x}$ be the second fundamental form of $\partial\Omega$ at $x$, i.e. $A$ is the matrix of…

We give upper bounds on the genus of a curve with general moduli assuming that it can be embedded in a projective nonsingular surface $Y$ so that $\dim(|C|) > 0$. We find such bounds for all types of surfaces of intermediate Kodaira…

In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description…

We study constant mean curvature surfaces in the three-dimensional Heisenberg group. We prove that a constant mean curvature surface in a neighborhood of non-umbilic point is described by some solution of a sinh-Gordon equation subject to a…

Error estimates are proved for an evolving surface finite element semi-discretization for anisotropic mean curvature flow of closed surfaces. For the geometric surface flow, a system coupling the anisotropic evolution law to parabolic…

For a smooth irreducible curve $C$, its second gonality $d_2$ is defined to be the minimum integer $d$ such that $C$ admits a linear series $g_d^2$. In this paper, we compute the second gonality of a smooth aCM curve $C$ lying on a smooth…

We find the explicit local equations of biconservative surfaces with non-constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, when the gradient of the mean curvature function is a principal…

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with…

We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that…

We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.

We establish the global $C^2$-estimates for the modified $\sigma_2$ curvature equation with prescribed boundary mean curvature, and particularly, the local boundary $C^2$ estimates on three-manifolds.

We study the variational behavior of the total inverse mean curvature of curves with prescribed boundary in the half-plane. We characterize the existence of critical points with prescribed area. We show that such critical points are…

It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the…

Determining the positions of lattice defects on elastic surfaces with Gaussian curvature is a non-trivial task of mechanical energy optimization, particularly for surfaces with boundaries. We introduce a simple way to predict the onset of…