Minimal surfaces and Schwarz lemma

2017/08/06 by David Kalaj, Kalaj, David
Engineering · Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Point processes and geometric inequalities #Polymer Science and Applications

paper · pdf · doi:10.48550/arxiv.1708.01848

openalex publication_date 2017/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If F:D→ Σ is a conformal harmonic parameterization of a minimal disk Σ, where D is the unit disk and |Σ|=πR², then |F_x(z)|(1-|z|²)≤ R. If for some z the previous inequality is equality, then the surface is an affine disk, and F is linear up to a Möbius transformation of the unit disk.

Related

• Ahlfors-Weill Extensions for a Class of Minimal Surfaces
• The Schwarzian derivative and the degree of a classical minimal surface
• The Schwarz lemma for holomorphic and minimal disks at the boundary
• Minimal Surface Linear Combinatoin Theorem
• Canonical Weierstrass representations for minimal surfaces in Euclidean\n 4-space