Related papers: Structure theorems for embedded disks with mean cu…

We construct a sequence of embedded minimal disks in a ball where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper.

We prove that a closed immersed plane curve with total curvature $2\pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct…

We are concerned with two interrelated problems: smoothability of connection 1-forms with low regularity on bundles with prescribed smooth curvature 2-forms, and existence of isometric immersions with low regularity. We first show that if…

We prove that $m$-dimensional Lipschitz graphs in any codimension with $C^{1,\alpha}$ boundary and anisotropic mean curvature bounded in $L^p$, $p > m$, are regular at every boundary point with density bounded above by $1/2 +\sigma$,…

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the…

Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $\rho^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$,…

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski…

This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The…

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4\pi must be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if…

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary…

Self-shrinkers are the special solutions of mean curvature flow in $\mathbf{R}^{n+1}$ that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding-Minicozzi is a…

We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension…

A graph is nearly embedded in a surface if it consists of graph $G_0$ that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly…

We first consider immersions on compact manifolds with uniform $L^p$-bounds on the second fundamental form and uniformly bounded volume. We show compactness in arbitrary dimension and codimension, generalizing a classical result of J.…

We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature…

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm…

Motivated by the large ammount of results obtained for minimal and positive constant mean curvature surfaces in several ambient spaces, the aim of this paper is to obtain half-space theorems for properly immersed surfaces in $\mathbb{R}^3$…

The disk embedding lemma is a technique underlying the topological classification results in 4-manifold topology for good fundamental groups. The purpose of this paper is to develop new tools for disk embedding that work up to s-cobordism,…

In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these…