Related papers: Constant Mean Curvature Surfaces in Homology Class…
We show that total generalized mean curvatures of hypersurfaces with positive reach in Riemannian manifolds, and convex bodies in Cartan-Hadamard spaces, are continuous with respect to Hausdorff distance.
We study global aspects of the mean curvature flow of non-separating hypersurfaces $S$ in closed manifolds. For instance, if $S$ has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal…
We classify complete orientable hypersurfaces of constant isotropic curvature in space forms. We show that such a hypersurface has constant mean curvature only if it is an isoparametric hypersurface, and that it is minimal if and only if it…
This paper extends the results of [GLS24], where the existence of a constant harmonic mean curvature foliation was established in the setting of a 3-dimensional asymptotically Schwarzschild manifold. Here, we generalize this construction to…
In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…
We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if…
One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and…
A sequence of constant mean curvature surfaces $\Sigma_j$ with mean curvature $H_j \to \infty$ in a three-dimensional manifold $M$ condenses to a compact and connected graph $\Gamma$ consisting of a finite union of curves if $\Sigma_j$ is…
We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known…
In this paper we study constant angle surfaces in Euclidean 3-space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the…
We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.
We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out…
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…
The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed…
We prove that, given $|H|<1$, a generic simple closed curve embedded in the asymptotic boundary of $\mathbb{H}^3$ (with respect to the supremum metric) bounds more than one complete surface embedded in $\mathbb{H}^3$ which has constant mean…
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.
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…
We establish the existence of hypersurfaces with constant mean curvature and a prescribed boundary in Euclidean space, represented as radial graphs over domains of the unit sphere. Under the assumptions that the mean curvature of the…
A gap in the proof prevents us to show that surfaces with constant mean curvature closed to 1/2 in H2 X R and having boundary with curvature greater than one, contained in a horizontal section P of H2 X R are topological disks, provided…
We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean…