On harmonic and biharmonic maps from gradient Ricci solitons

TL;DR

This paper investigates harmonic and biharmonic maps originating from gradient Ricci solitons, establishing conditions that simplify their behavior and demonstrating that certain finite energy biharmonic maps are necessarily harmonic.

Contribution

It provides new analytic and geometric criteria for harmonic and biharmonic maps from gradient Ricci solitons, including the case of the two-dimensional cigar soliton.

Findings

Harmonic maps are constant under specific conditions.

Biharmonic maps are forced to be harmonic in certain cases.

Finite energy biharmonic maps from the cigar soliton are harmonic.

Abstract

We study harmonic and biharmonic maps from gradient Ricci solitons. We derive a number of analytic and geometric conditions under which harmonic maps are constant and which force biharmonic maps to be harmonic. In particular, we show that biharmonic maps of finite energy from the two-dimensional cigar soliton must be harmonic.