Related papers: Closed self-shrinking surfaces in $\mathbb{R}^3$ v…

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round…

In this paper we solve the Bj\"orling problem for the class of immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with…

We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we…

In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in $\mathbb{R}^{n+1}$. First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the…

We extend the theory of complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature to the wider class of elliptic special Weingarten surfaces of finite total curvature; in particular, we extend the seminal works of L. Jorge and…

This is an investigation into a classification of embeddings of a surface in Euclidean $3$-space. Specifically, we consider $\mathbb{R}^3$ as having the product structure $\mathbb{R}^2 \times \mathbb{R}$ and let $\pi:\mathbb{R}^2 \times…

In this note, we describe the possible singularities on a stable surface which is in the boundary of the moduli space of surfaces isogenous to a product. Then we use the $\mathbb Q$-Gorenstein deformation theory to get some connected…

We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its…

We consider 2-dimensional orientable self-shrinkers $\Sigma$ for the Mean Curvature Flow of polynomial volume growth immersed in $\mathbb R^n$. We look at closed one forms minimizing the norm $\int_\Sigma \eterm |\omega|^2$ in their…

A novel class of discrete integrable surfaces is recorded. This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, `linear' Weingarten, Guichard and Petot surfaces. Moreover,…

We apply the loop group method developed by Zakharov-Shabat, Terng-Uhlenbeck and Toda to the study of symmetries of pseudospherical surfaces in R^3. In this paper (part I) we consider the general theory, while in a second paper (part II) we…

We prove that singular minimal surfaces with constant Gauss curvature are planes, spheres and cylindrical surfaces. We also classify all singular minimal surfaces with a constant principal curvature and singular minimal surfaces with…

We verify that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfying $0\leq |A|^2-n\leq\frac{n}{22}$, then $|A|^2\equiv n$ and $M$ is a Clifford torus. Moreover, we…

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similar solutions are the…

Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic…

We construct a new class of complete constant mean curvature surfaces in R^3. These are geometrically different than the surfaces constructed by Kapouleas' gluing technique. These are obtained by piecing together half-Delaunay surfaces to…

In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $\Sigma$ is a closed minimal surface embedded in a Riemannian three-manifold…

The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case.…

A translation surface in the three-dimensional sphere $\mathbb{S}^3$ is a surface generated by the quaternionic product of two curves, called generating curves. In this paper, we present rigidity results for such surfaces. We introduce an…

Minimal area surfaces in AdS$_3$ ending on a given curve at the boundary are dual to planar Wilson loops in N=4 SYM. In previous work it was shown that the problem of finding such surfaces can be recast as the one of finding an appropriate…