Existence of a constant-mean-curvature hypertorus in \(S^4\) via computer assistance

TL;DR

This paper proves the existence of a new embedded constant mean curvature hypertorus in the 4-sphere using computer-assisted rigorous numerical methods, combining the Taylor method with the Poincare-Miranda theorem.

Contribution

It introduces a novel computer-assisted approach to establish the existence of complex geometric structures in higher-dimensional spheres.

Findings

Existence of a new embedded CMC hypertorus in S^4 confirmed.

Method combines rational arithmetic Taylor method with topological theorems.

Provides rigorous numerical proof of the hypertorus's existence.

Abstract

The round Taylor method uses rational arithmetic, allowing control of both round-off and truncation errors in approximating solutions of differential equations. In this paper, we employ this method together with the Poincare-Miranda theorem to prove the existence of a new embedded constant mean curvature (CMC) hypertorus in the unit four dimensional sphere