Infinity-Minimal Submanifolds

TL;DR

This paper explores the variational principles behind infinity-harmonic maps, introduces the concept of infinity-minimal maps, and characterizes minimal surfaces in 3-space through these new notions, establishing key properties and principles.

Contribution

It introduces infinity-minimal maps as solutions to the infinity-Laplacian and characterizes minimal surfaces via isothermal immersions with infinity-minimal area.

Findings

Infinity-minimal maps solve the infinity-Laplacian.

A maximum principle for |Du| is established.

Minimal surfaces are characterized by isothermal infinity-harmonic maps.

Abstract

We identify the Variational Principle governing inifinity-Harmonic maps, that is solutions to the Infinity-Laplacian. The system was first derived in the limit of the p-Laplacian as p->inifinity in [K2] and is recently studied in [K3]. Here we show that it is the "Euler-Lagrange PDE" of vector-valued Calculus of Variations in L-inifinity for the L-inifinity norm of the gradient. We introduce the notion of inifinity-Minimal Maps, whch are Rank-One Absolute Minimals of with inifinity-Minimal Area" of the range submanifold and prove they solve the inifinity-laplacian. The converse is true for immersions. We also establish a maximum principle for |Du| for solutions. We further characterize minimal surfaces of 3-space as those locally parameterizable by isothermal immersions with inifinity-minimal area and show that isothermal inifinity-Harmonic maps are rigid.