arXiv research

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

This paper solves minimal surface equations near Hardt-Simon foliations.

An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for minimal surfaces. Here we explain not only how this method works but also how we c…

We develop Teichmuller theoretical methods to construct new minimal surfaces in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in $\BE^3$. We exhibit this method on an interesting class of minimal surfaces which are likely to be embedded, and have a low degree Gaußmap for their genus; the (Weier…

Unstable minimal surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for minimal surfaces in the Euclidean spacce was presented in \cite{s3}. We…

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…

New method glues Scherk surfaces into minimal surfaces, limiting possible outcomes.

New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.

We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to …

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…

In this paper, we employ the loop group method to study the construction of minimal Lagrangian surfaces in the complex projective plane for which the surface is contractible. We present several new classes of minimal Lagrangian surfaces in $\mathbb{C} P^2$.

Complete minimal surfaces with any genus found in a specific 3-manifold.

The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…

New minimal surface doublings of Clifford Torus improve bounds on minimal surfaces in S^3.

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.

The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.

The paper studies minimal submanifolds with specific curvature properties in Euclidean space.

DPW method reconstructs minimal and symmetric CMC surfaces in 3-sphere.

Unified representation for minimal and constant mean curvature surfaces.

Minimal annuli constructed in PSL2 via variational method.

Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.

The paper studies minimal surfaces in 3D spheres and balls, confirming conjectures and identifying new surfaces.

Extends symmetry and rigidity to surfaces with soap film-like singularities.

Study on existence and structure of P-area surfaces in Heisenberg group.

We construct minimal surfaces in hyperbolic and anti-de Sitter 3-space with the topology of a $n$-punctured sphere by loop group factorization methods. The end behavior of the surfaces is based on the asymptotics of Delaunay-type surfaces, i.e., rotational symmetric minimal cylinders. The minimal surfaces in $\mathrm{H…

The study bounds the topology of free boundary minimal surfaces in 3D manifolds.

Develops a new method for minimal Lagrangian surfaces in complex quadrics.

The paper constructs free boundary minimal surfaces in product spaces using eigenvalue methods.

Study nonlocal minimal surfaces for minimal surfaces in 3D manifolds.

We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass-representation.

We study symmetric minimal surfaces in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will discuss how to construct minimal surfaces in $\mathrm{Nil}_3$ with non-trivial topology. Moreover, we will classif…

In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.

We construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.

Proves existence of minimal surfaces of arbitrary genus with two ends.

The study proves theorems about minimal and H-surfaces in 3D space.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …

Proves Willmore conjecture for surfaces with specific symmetries.

Constructs cmc doublings of minimal surfaces via min-max theory.

Abstract framework for two meromorphic forms on punctured surfaces.

We introduce a new technique to solve period problems on minimal surfaces called limit-method. If a family of surfaces has Weierstrass-data converging to the data of a known example, and this presents a transversal solution of periods, then the original family contains a sub-family with closed periods.

We establish parabolicity and quadratic area growth for minimal surfaces-with-boundary contained in regions of R^3 which are within a sub-logarithmic factor of the exterior of a cone. Unlike previous work showing that these two properties hold for minimal surfaces-with-boundary contained between two catenoids, we do no…

This paper embeds surfaces in 3D spheres and balls with minimal area.

Machine learning predicts minimal surfaces for knots, supporting a conjecture.

Using Traizet's regeneration method, we prove that for each positive integer n there is a family of embedded, doubly periodic minimal surfaces with parallel ends in Euclidean space of genus 2n-1 and 4 ends in the quotient by the maximal group of translations. The genus 2n-1 family converges smoothly to 2n copies of Sch…

Let $α$ be a polygonal Jordan curve in $\bfR^3$. We show that if $α$ satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary $α$ is unique and is a smooth graph. As our conditions on $α$ are not included amongst previously known conditions for embeddedness, we are enlarging the set…

We investigate solutions to the minimal surface problem with Dirichlet boundary conditions in the roto-translation group equipped with a subRiemannian metric. By work of G. Citti and A. Sarti, such solutions are amodal completions of occluded visual data when using a model of the first layer of the visual cortex. Using…

It is pointed out that despite of the non-linearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.