A duality result between the minimal surface equation and the maximal surface equation

Luis J Alías,Bennett Palmer

doi:10.1590/s0001-37652001000200002

Luis J Alías, Bennett Palmer

Open Access

https://doi.org/10.1590/s0001-37652001000200002

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Abstract

In this note we show how classical Bernstein's theorem on minimal surfaces in the Euclidean space can be seen as a consequence of Calabi-Bernstein's theorem on maximal surfaces in the Lorentz-Minkowski space (and viceversa). This follows from a simple but nice duality between solutions to their corresponding differential equations.

Highlights

A minimal surface in Euclidean space R3 is a surface with zero mean curvature
A maximal surface in the Lorentz-Minkowski space L3 is a spacelike surface with zero mean curvature
In this note we show how classical Bernstein’s theorem on minimal surfaces in the Euclidean space R3 can be seen as a consequence of Calabi-Bernstein’s theorem on maximal surfaces in the Lorentz-Minkowski space L3

Summary

Introduction

A minimal surface in Euclidean space R3 is a surface with zero mean curvature. Bernstein (19151917) proved that the planes are the only minimal entire graphs in R3.Theorem 1. (Bernstein’s theorem). A minimal surface in Euclidean space R3 is a surface with zero mean curvature. Bernstein (19151917) proved that the planes are the only minimal entire graphs in R3. The only entire solutions to the minimal surface equation

Results

Conclusion