A proof by foliation that Lawson's cones are $A_{\Phi}$-minimizing

TL;DR

This paper proves that cones over products of spheres minimize certain elliptic functionals using foliation methods and explores the asymptotic behavior of these foliations, leading to conjectures on solutions to minimal surface equations.

Contribution

It introduces a foliation-based proof that Lawson's cones are $A_{ ext{Phi}}$-minimizing and analyzes their behavior at infinity, suggesting new conjectures.

Findings

Lawson's cones over $ ext{S}^k imes ext{S}^l$ are $A_{ ext{Phi}}$-minimizing.

Foliation approach provides a new proof technique.

Asymptotic analysis leads to conjectures on nonlinear minimal surface solutions.

Abstract

We give a proof by foliation that the cones over $\mathbb{S}^k \times \mathbb{S}^l$ minimize parametric elliptic functionals for each $k,\,l \geq 1$. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.