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arXiv · 1708.01848

Minimal surfaces and Schwarz lemma

David Kalaj
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Abstract

We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma$, where $\mathbf{D}$ is the unit disk and $|\Sigma|=\pi R^2$, then $|F_x(z)|(1-|z|^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine disk, and $F$ is linear up to a M\"obius transformation of the unit disk.

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David Kalaj. 2017-08-06. Minimal surfaces and Schwarz lemma. https://arxiv.org/abs/1708.01848

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