Related papers: Capillary surfaces in a cone
We show that constant mean curvature hypersurfaces in $\mathbb H^n\times\mathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces…
On a compact surface, we prove existence and uniqueness of the conformal metric whose curvature is prescribed by a negative function away from finitely many points where the metric has prescribed angles presenting cusps or conical…
We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M\times\R$ where $M^n$ is a Riemannian submanifold in $\R^{n+1}$. The main purpose is to establish a priori gradient estimates for…
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…
Given a function $\mathcal{H} \in C^1(\mathbb{S}^2)$, an $\mathcal{H}$-surface $\Sigma$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_\Sigma$ satisfies $H_\Sigma = \mathcal{H} \circ \eta$, where $\eta$ is the…
Liquid in grooved capillaries, made by e.g. inserting a plate in a cylindrical tube, exhibits unusual spreading and flow properties. One example is capillary rise, where a long, upward tongue on top of the usual meniscus has been observed…
We study the formation of images in a reflective sphere in three configurations using caustics of the field of light rays. The optical wavefront emerging from a source point reaching a subject following passage through the optical system…
In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar…
We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also…
For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes…
We prove that a surface in Euclidean $3$-space has Maslovian normal bundle if and only if it is a part of a round sphere, a circular cylinder, or a circular cone.
The dynamic conformal transformation method has been generalized for the first time to numerically simulate the capillary wave turbulence of a liquid surface in the plane symmetric anisotropic geometry. The model is strongly nonlinear and…
We consider regular surfaces $M$ that are given as the zeros of a polynomial function $p:R^3\rightarrow R$, where the gradient of $p$ vanishes nowhere. We assume that $M$ has non-zero mean curvature and prove that there exist only two…
Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the…
We study the soap film capillarity problem, in which soap films are modeled as sets of least perimeter among those having prescribed (small) volume and satisfying a topological spanning condition. When the given boundary is the closed…
The turbulence of capillary waves on the surface of a ferrofluid with a high permeability in a horizontal magnetic field is considered in the framework of a one-dimensional weakly nonlinear model. In the limit of a strong magnetic field,…
The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of…
In this article, we construct complete embedded constant mean curvature surfaces in $\mb{R}^3$ with freely prescribed genus and any number of ends greater than or equal to four. Heuristically, the surfaces are obtained by resolving finitely…
Consider a mean curvature flow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the flow is graphical inside the cylinder of half the radius. Here we prove a…
A constant angle surface in Minkowski space is a spacelike surface whose unit normal vector field makes a constant hyperbolic angle with a fixed timelike vector. In this work we study and classify these surfaces. In particular, we show that…