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Minimal surfaces in Euclidean spaces by way of complex analysis

Franc Forstneric

Abstract

This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portorož on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of conformal minimal surfaces in Euclidean spaces. New results concern approximation, interpolation, and general position properties of minimal surfaces, existence of minimal surfaces with a given Gauss map, and the Calabi-Yau problem for minimal surfaces. To be accessible to a wide audience, the article includes a self-contained elementary introduction to the theory of minimal surfaces in Euclidean spaces.

Minimal surfaces in Euclidean spaces by way of complex analysis

Abstract

This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portorož on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of conformal minimal surfaces in Euclidean spaces. New results concern approximation, interpolation, and general position properties of minimal surfaces, existence of minimal surfaces with a given Gauss map, and the Calabi-Yau problem for minimal surfaces. To be accessible to a wide audience, the article includes a self-contained elementary introduction to the theory of minimal surfaces in Euclidean spaces.

Paper Structure

This paper contains 12 sections, 5 theorems, 75 equations.

Table of Contents

  1. Minimal surfaces: a link between mathematics, science, engineering, and art
  2. An elementary introduction to minimal surfaces
  3. Conformal maps and conformal structures on surfaces.
  4. First variation of area and energy
  5. Characterization of minimality by vanishing mean curvature
  6. The Enneper--Weierstrass representation
  7. Holomorphic null curves
  8. A survey of new results
  9. Approximation, interpolation, and general position theorems
  10. Topological structure of spaces of minimal surfaces
  11. The Gauss map of a conformal minimal surface
  12. The Calabi--Yau problem

Key Result

Theorem 2.1

Let $D$ be a relatively compact domain in $\mathbb{R}^2$ with piecewise smooth boundary. A smooth conformal immersion $X:\overline D\to\mathbb{R}^n$$(n\ge 3)$ is a stationary point of the area functional eq:area if and only if $X$ is harmonic: $\Delta X=0$.

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Definition 2.5
  • Theorem 2.6: The Enneper-Weierstrass representation
  • Remark : The first homology group
  • Example 2.7: Catenoid
  • ...and 5 more