Related papers: Bubbletons are not embedded
We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov…
We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.
Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely…
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…
In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet $\mathbb S^1\times\mathbb R^3$ , and we derive an asymptotic profile for a neighborhood of singularity.
We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on…
For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the…
We start the investigation of immersions $\Psi$ of a simply connected domain $D$ into three dimensional Euclidean space $R^3$, which have constant mean curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which leave…
.The physical properties of an ensemble of tightly packed particles like bubbles, drops or solid grains are controlled by their interactions. For the case of bubbles and drops it has recently been shown theoretically and computationally…
Say S is a compact three-manifold with non-positive Yamabe invariant. We prove that in any long time constant mean curvature Einstein flow over S, having bounded C^{\alpha} space-time curvature at the cosmological scale, the reduced volume…
We prove that horospheres, hyperspheres and hyperplanes in a hyperbolic space H n , n $\ge$ 3, admit no perturbations with compact support which increase their mean curvature. is is an extension of the analogous result in the Euclidean…
This is a continuation of the authors' earlier work on deformations of cuspidal edges. We give a representation formula for swallowtails in the Euclidean 3-space. Using this, we investigate map germs of generic swallowtails in 3-dimensional…
Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean…
In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in Kapouleas (1990) by adopting the more precise and powerful version of the methodology which…
We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are…
We show that the mean curvature flow of a generic closed surface in $\mathbb{R}^3$ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical…
We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.
We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then…
In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $\Sigma$ is a complete, properly…
We prove that there are no minimal hypersurfaces properly immersed in any region of the Euclidean space bounded by unstable minimal cones. We also prove the analogous result for $r$-minimal hypersurfaces.