A construction of conformal-harmonic maps

TL;DR

This paper establishes the existence of conformal harmonic maps from 4-dimensional conformal manifolds to Riemannian manifolds using a geometric flow approach, extending classical harmonic map results to a conformally invariant setting.

Contribution

It introduces a general existence theorem for conformal harmonic maps, analogous to the Eells-Sampson theorem, employing a geometric flow method and building on recent geometric analysis results.

Findings

Proves existence of conformal harmonic maps under broad conditions

Develops a geometric flow approach for fourth order conformally invariant equations

Extends classical harmonic map theory to conformal geometry context

Abstract

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky-Viaclovsky and Lamm.