Related papers: Minimal surfaces near Hardt-Simon foliations

We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact…

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in…

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct…

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the…

We establish a general min-max type theorem that produces minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature. An important intermediate step is to show that, in a generic metric with positive Ricci…

In the work \cite{Laredo} the author shows that every hypersurface in Euclidean space is locally associated to the unit sphere by a sphere congruence, whose radius function $R$ is a geometric invariant of hypersurface. In this paper we…

We show that min-max minimal hypersurfaces can be localized. As a consequence, we obtain the sharp generalization to complete manifolds of the famous Almgren-Pitts min-max theorem in closed manifolds. We use this result to prove the…

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of…

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…

In this paper we introduce the Chern minimal surface in Hermitian surfaces by using the Chern connection, and we show that it only has isolated complex and anticomplex points for a generic one (neither holomorphic nor antiholomorphic). For…

In this paper, we construct a family of mean curvature flow which converges to an area minimizing, strictly stable hypercone $\mC$ after type I rescaling, and converges to the Hardt-Simon foliation of the cone after a type II rescaling…

We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D…

We prove that, given a compact Riemann surface $\Sigma$ and disjoint finite sets $\varnothing\neq E\subset\Sigma$ and $\Lambda\subset\Sigma$, every map $\Lambda \to \mathbb{R}^3$ extends to a complete conformal minimal immersion…

In this paper, we first classify singular fibers of proper $C^\infty$ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and…

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…

In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete,…

Given an open Riemann surface $M$, we show that the branch points and the complete ends of finite total curvature of a conformal minimal surface $M\to{\mathbb R}^n$, $n\ge 3$, can be removed by an isotopy through such surfaces. The…

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we construct a quasi-Fuchsian manifold which…

We compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on projective hypersurfaces of fixed degree. In particular, we introduce a…

Minimal and CMC surfaces in $S^3$ can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which…