Related papers: Capillary surfaces in a cone

We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for $G^1$ smoothness are given, which, up to a certain degenerate case, are also…

In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We…

In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles and that satisfy a Weingarten condition of type $a H+b K=c$, where $a,b$ and $c$ are constant and $H$ and $K$ denote the mean curvature and the Gauss…

We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius…

We consider capillary surfaces that are constructed by bounded generating curves. This class of surfaces includes radially symmetric and lower dimensional fluid-fluid interfaces. We use the arc-length representation of the differential…

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is…

A projectability result is proved for surfaces of prescribed mean curvature (shortly called $H$-surfaces) spanned in a partially free boundary configuration. Hereby, the $H$-surface is allowed to meet the support surface along its free…

Given a $C^1$ function $\mathcal{H}$ defined in the unit sphere $\mathbb{S}^2$, an $\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\mathcal{H}(N_p)$, $p\in M$, where…

Solving a long-standing open question in convex geometry, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of…

It is proved that a generic simple, closed, piecewise regular curve in space can be the boundary of only finitely many developable surfaces with nonvanishing mean curvature. The relevance of this result in the context of the dynamics of…

The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C^2$ norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant…

We present a collection of easily stated open problems in the theory of compact constant mean curvature surfaces with boundary. We also survey the current status of answering them.

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…

Cone spherical surfaces are orientable Riemannian surfaces with constant curvature one and a finite set of conical singularities. A subset of these surfaces, referred to as dihedral surfaces, is characterized by their monodromy groups,…

If a droplet is placed on a substrate with a conical shape it spontaneously starts to spread in the direction of a growing fibre radius. We describe this capillary spreading dynamics by developing a lubrication approximation on a cone and…

The dynamics of saturated vapor between two intersecting walls is examined. It is shown that, if the angle $\phi$ between the walls is sufficiently small, the vapor becomes unstable, and spontaneous condensation occurs in the corner,…

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of…

We solve the problem of prescribing different types of curvatures (principal, mean or Gaussian) on rotational surfaces in terms of arbitrary continuous functions depending on the distance from the surface to the axis of revolution. In this…