Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields

TL;DR

This paper classifies biharmonic hypersurfaces in spheres and projective spaces, addresses Chen's conjecture for biharmonic maps, and explores bi-Yang-Mills fields, revealing their isolation phenomena.

Contribution

It provides comprehensive classifications of biharmonic hypersurfaces and extends the analysis to bi-Yang-Mills fields, including their isolation phenomena.

Findings

Classified all biharmonic isoparametric hypersurfaces in the sphere.

Solved cases of Chen's conjecture and related conjectures for biharmonic maps.

Discovered the isolation phenomena of bi-Yang-Mills fields.

Abstract

Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a manifold of non positive curvature are given. Gauge field analogue is shown, and the isolation phenomena of bi-Yang-Mills fields are obtained.