Minimal hypertori in the four-dimensional sphere

TL;DR

This paper proves the existence of minimal hypertori and hyperspheres in the four-dimensional sphere, providing new solutions to longstanding geometric conjectures and demonstrating the richness of minimal submanifolds in higher dimensions.

Contribution

It establishes the existence of minimal hypertori and hyperspheres in the 4-sphere, solving Chern's spherical Bernstein conjecture in dimensions four and six.

Findings

Existence of minimally embedded hypertori in the 4-sphere

Infinitely many non-isometric minimally immersed hypertori

Infinitely many non-isometric minimally embedded hyperspheres

Abstract

We prove that the four-dimensional round sphere contains a minimally embedded hypertorus, as well as infinitely many, pairwise non-isometric, immersed ones. Our analysis also yields infinitely many, pairwise non-isometric, minimally embedded hyperspheres and thus provides a self-contained solution to Chern's spherical Bernstein conjecture in dimensions four and six.