In early work, I devised methods to construct harmonic maps by exploiting symmetries given by isoparametric functions.  These so-called reduction techniques reduce the number of variables leading to a more accessible set of equations to solve.  Book

Together with John C. Wood, in a series of papers we characterize harmonic morphisms from 3-manifolds to a surface.   This work involves techniques from several domains: algebra through polynomial mappings, analysis through polar sets and removable singularities of solutions to differential equations, topology through foliation theory and of course geometry.  The work culminated in a London Mathematical Society Monograph Book.   Harmonic morphisms are mappings which preserve solutions to Laplace's equation.  They are characterized as mappings which are both harmonic and semi-conformal.   The latter condition is conformally invariant and becomes interesting in its own right.  

Semi-conformal mappings to a surface are defined by pairs of conjugate functions, that is functions whose gradients are everywhere orthogonal and of the same length.  In dimension 2, functions which locally admit a conjugate are characterized as the harmonic functions, but what happens in higher dimensions.  In joint work with Mike Eastwood (Annals de l'Institut Fourier 2015), we show that in 3-dimensional Euclidean space, the functions which admit a conjugate are characterized as those which satisfy a 2nd order differential inequality and three 3rd order differential equations (all conformally invariant). 

Continuing the theme of conformal geometry, in joint work with Ali Fardoun and Rachid Regbaoui, we study by an evolution equation the prescribed curvature problem, first for the Gauss curvature on a surface (Ann. Scuola Normale Sup. Pisa 2004), and then for Q-curvature in dimension 4 (Calculus of Variations and PDEs 2006).  This latter work required the positivity of the Paneitz operator.  We were able to remove this and extend the work to general even dimensional manifolds using a min-max method in Journal of Geometry and Physics 2009. 

The Ricci flow is a topic at the heart of differential geometry at the present time.  In joint work with my doctoral student Laurent Danielo, we found the first known examples of non-gradient Ricci solitons (also found independently by John Lott), see Journal  reine  angew. Mathematik (2007).  These objects are fundamental to the understanding of the long term behaviour of solutions to the Ricci flow. 

The study of biharmonic maps is intimately related to mean curvature.  In joint work with Ali Fardoun and Seddik Ouakkas, we explore relations between biharmonic semi-conformal maps and the means curvature of their fibres (Ann. Global Analysis and Geometry 2008), as well as Liouville-type theorems for biharmonic maps (Advances in Calculus of Variations 2010).