- Differential geometry and geometric analysis.
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).