相关论文: A connection between minimal surfaces and the two-…

A surface in Euclidean space $\r^3$ is said to be an $\alpha$-stationary surface if it is a critical point of the energy $\int_\Sigma|p|^\alpha$, where $\alpha\in\r$. We prove that all ruled $\alpha$-stationary surfaces are vector planes…

Consider the energy $E_\alpha[\Sigma]=\int_\Sigma |p|^\alpha\, d\Sigma$, where $\Sigma$ is a surface in Euclidean space $\r^3$ and $\alpha\in\r$. We prove that planes and spheres are the only stationary surfaces for $E_\alpha$ with constant…

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature $H$ at each point $x$ satisfies $2H(x)=a r^2+b$, where $r$ is the distance from $x$ to a fixed straight-line $L$, and $a$ and $b$ are constants.…

We investigate axisymmetric surfaces in Euclidean space that are stationary for the energy $E_\alpha=\int_\Sigma |p|^\alpha\, d\Sigma$. By using a phase plane analysis, we classify these surfaces when they intersect orthogonally the…

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general…

Let $\alpha\in\r$ and let $\vec{v}\in\r^3$ be a unit vector. A singular minimal surface $\Sigma$ in Euclidean space is a surface $\Sigma$ whose mean curvature $H$ satisfies $H=\alpha\frac{\langle N,\vec{v}\rangle}{\langle…

Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product $\Sigma_1\times\Sigma_2$ of two complete orientable Riemannian surfaces of…

In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we…

A translation surface of Euclidean space $\r^3$ is the sum of two regular curves $\alpha$ and $\beta$, called the generating curves. In this paper we classify the minimal translation surfaces of $\r^3$ and we give a method of construction…

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…

In this paper, we extend the notion of stationary curves with respect to the moment of inertia from a point $N$ in the Euclidean plane $\mathbb{R}^2$ to the case that the ambient space is either the hyperbolic plane $\mathbb{H}^2$ or the…

The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system…

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $\phi(x,y,z)=\alpha x+\beta y+\gamma y$, where $\alpha,\beta,\gamma$ are real numbers not all zero. We prove that if a surface is $\phi$-minimal…

We consider Lie minimal surfaces, the critical points of the simplest Lie sphere invariant energy, in Riemannian space forms. These surfaces can be characterized via their Euler-Lagrange equations, which take the form of differential…

For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0…

We study the 2-systole on compact K\"ahler surfaces of positive scalar curvature. For any such surface $(X,\omega)$, we prove the sharp estimate $\min_X S(\omega)\cdot\operatorname{sys}_2(\omega)\le 12\pi$, with equality if and only if…

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$.…

In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…

We give an explicit construction of any simply-connected superconformal surface $\phi\colon M^2\to \R^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h\colon M^2\to\R^4$. That $\phi$ is superconformal means that…

We investigate equilibrium configurations for surface energies which contain the squared $L^2$ norm of the difference of the mean curvature H and the spontaneous curvature $c_o$ coupled with the elastic energy of the boundary curve, which…