Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMinimal surfaces in Euclidean spaces by way of complex analysis
102
0
0.0
(
0
)
Authors
Franc Forstneric
Ask ChatGPT about the research
No Arabic abstract
This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portorov{z} 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.
rate research
Read More
We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc $mathbb D$ in $mathbb C$ into the unit ball $mathbb B^n$ in $mathbb R^n$, $nge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $nge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $mathbb Dto mathbb B^n$. This implies that conformal harmonic immersions $M to mathbb B^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincar$mathrm{e}$ metric on $M$ and the Cayley-Klein metric on the ball $mathbb B^n$, and the extremal maps are precisely the conformal embeddings of the disc $mathbb D$ onto affine discs in $mathbb B^n$. By using these results, we lay the foundations of the hyperbolicity theory for domains in $mathbb R^n$ based on minimal surfaces.
We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of branch points. On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fary-Milnors theorem in the CAT(0) setting.
In this article, we interpolate a given real analytic spacelike curve $a$ in Lorentz-Minkowski space $mathbb{L}^3$ to another real analytic spacelike curve $c$, which is close enough to $a$ in a certain sense, by a maximal surface using inverse function theorem for Banach spaces. Using the same method we also interpolate a given real analytic curve $a$ in Euclidean space $mathbb{E}^3$ to another real analytic curve $c$, which is close enough to $a$ in a certain sense, by a minimal surface. The Bjorling problem and Schwartzs solution to it play an important role.
We consider immersions of a Riemann surface into a manifold with $G_2$-holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
