Classical Theory Of Minimal Surfaces Research Articles

Stable cones in the thin one-phase problem

abstract: The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $n\geq3$ is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of ``large solutions'' for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions $n\le 5$ is one-dimensional, \emph{independently} of the parameter $s\in (0,1)$.

NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY

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 Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$ , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

The classical theory of minimal surfaces

We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid and the one-parameter family fRtgt2(0;1) of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in R 3 ; the proof of this result depends primarily on work of Colding

Properly embedded minimal planar domains with infinite topology are Riemann minimal examples

These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M R 3 with innite topology (equivalently, with an innite number of ends). This

Geometric results in classical minimal surface theory

Geometric results in classical minimal surface theory

The Gauss map for Kählerian submanifolds of 𝑅ⁿ

We introduce a Gauss map for Kähler submanifolds of Euclidean space and study its geometry in relation to that of the given immersion. In particular we generalize a number of results of the classical theory of minimal surfaces in Euclidean space.

The topology of complete minimal surfaces of finite total Gaussian curvature

RECENTLY GREAT progress has been made in the classical theory of minimal surfaces in R3. For example, the proof of the embedding of the solution to Plateau’s problem for an extremal Jordan curve [lo] and the embedding of a solution to the free boundary value problem [ll], the proof of the bridge theorem [12], the regularity and finiteness of least area oriented surfaces bounding a smooth Jordan curve [5], the uniqueness and topological uniqueness of certain minimal surfaces 18, 9, 121, the proof that a complete stable minimal surface is a plane [l, 31, the existence of a collection of 4 Jordan curves which bound a continuous family of compact minimal surfaces [141, the existence of a complete minimal surface contained between two parallel planes [13] and a theorem [18] which states that if the Gauss map of a complete minimal surface misses 7 points on the ‘sphere, then the minimal surface is a plane. The aim of this work is to begin the exploration of the topological properties of complete minimal surfaces of finite total curvature. First we recall the fundamental classical result of Chern-Osserman [6] that states that such surfaces M are conformally equivalent to compact Riemann surfaces iii punctured in a finite number of points. Furthermore, they prove that the Gauss map on the surface M extends conformally to a. In particular, the theorem of Chern-Osserman implies that such surfaces have finite topological type with a finite number of topological ends and that the normal vectors at infinity on these ends are well defined. (Finite topological type means in this case that M is diffeomorphic to the interior of a compact surface with boundary.) Our first theorem gives a nice description of the topological placement in IX3 of a complete surface of finite topological type which has well defined normal vectors at infinity. We prove that the ends of such surfaces behave like the ends of the catenoid at infinity. More precisely, Theorem 1 shows that if f: M + R3 is an immersion of such a surface, then f is proper and the image f(M) viewed from infinity looks like a finite collection of flat planes (with multiplicity) that pass through the origin. For example, the catenoid viewed from infinity looks like two oppositely oriented copies of a plane passing through the origin. Theorem 1 has a natural generalization to submanifolds of arbitrary codimension in R”. This generalization which is Theorem 2 is discussed in

A survey of the geometric results in the classical theory of minimal surfaces

A survey of the geometric results in the classical theory of minimal surfaces

Conformal properties in classical minimal surface theory

This is a survey of recent developments in the classical theory of minimal surfaces in R3 with an emphasis on the conformal properties of these surfaces such as recurrence and parabolicity. We cover the maximum principle at infinity for properly immersed minimal surfaces in R3 and some new results on harmonic functions as they relate to the classical theory. We define and demonstrate the usefulness of universal superharmonic functions. We present the compactness and regularity theory of Colding and Minicozzi for limits of sequences of simply connected minimal surfaces and its application by Meeks and Rosenberg in their proof of the uniqueness of the helicoid. Finally, we discuss some recent deep results on the topology and on the index of the stability operator of properly embedded minimal surfaces and give an application of the classical Shiffman Jacobi function to the classification of minimal surfaces