Основні поняття
Approximating minimal surfaces using discrete methods.
Анотація
The article discusses solving the Björling problem by constructing minimal surfaces locally through discrete approximation. It explores the use of discrete conformal maps and Weierstrass representation to achieve this goal. The main focus is on determining suitable initial data from given real-analytic curves to approximate smooth minimal surfaces discretely. The process involves choosing appropriate data, constructing CR-mappings, and proving convergence to smooth counterparts. Various equations and methods are detailed throughout the article to support this approach.
- Introduction to Minimal Surfaces and the Björling Problem.
- From Björling Data to Cauchy Data for Weierstrass Representation.
- Construction of Rectangular Lattices and Discrete Holomorphic Functions.
- Cross-Ratio Evolution: Mappings from Cauchy Data.
- Derivation of a Discrete Evolution Equation for F ε.
- Consistency of the Discrete Evolution Equation (23).
- Detailed explanation of how discrete evolution equations are derived and their consistency with smooth counterparts.
Статистика
We prove that the approximation error is of the order of the square of the mesh size.
Given Bj¨orling data F0 and N0 and a point F0(t0) such that ˙F0(t0) is not parallel to ˙N0(t0), we can locally approximate the solution of the Bj¨orling problem F by discrete minimal surfaces Fm,n.