Related papers: Harmonic two-spheres in compact symmetric spaces, …

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $G$, whether $G$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group…

In their seminal 1981 article, Sacks-Uhlenbeck famously proved the existence of non-trivial harmonic 2-spheres in every closed Riemannian manifold with non-zero second homotopy group. Their arguments heavily rely on PDE techniques. The…

Making use of Murakami's classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic…

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of…

In [5], together with J. C. Wood, the authors gave a completely explicit formula for all harmonic maps from $2$-spheres to the unitary group $U(n)$ in terms of freely chosen meromorphic functions on $S^2$. The simplest harmonic maps are the…

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic…

We investigate in detail the connection between harmonic maps from Riemann surfaces into the unitary group $\U(n)$ and their Grassmannian models: these are families of shift-invariant subspaces of $L^2(S^1,\C^n)$. With the help of…

In this paper we consider sequences of $p$-harmonic maps, $p>2$, from a closed Riemann surface $\Sigma$ into the $n$-dimensional sphere $\mathbb{S}^n$ with uniform bounded energy. These are critical points of the energy $E_p(u)…

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then…

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ of finite uniton number, from a Riemann surface $M,$ compact or non-compact, into inner symmetric spaces of compact or non-compact type. As…

In the joint work of the author with Da Lio and Rivi\`ere (Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $p\searrow2$.…

This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from…

In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit…

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set,…

In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new…

We prove generic regularity and Uhlenbeck-type compactification theorems for the moduli spaces of PU(2)-monopoles. Generic regularity is NOT obtained in the usual way (by applying Sard theorem to a smooth parameterized moduli space), since…

We study biharmonic maps and f-biharmonic maps from a round sphere $(S^2, g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres $(S^2, f^{-1}g_0)$. We proved that for rotationally symmetric maps between rotationally…

We extend many known results for harmonic maps from the 2-sphere into a Grassmannian to harmonic maps of finite uniton number from an arbitrary Riemann surface. Our method relies on a new theory of nilpotent cycles arising from the diagrams…

The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group $Sp(2)$. It is shown that this oscillator system contains the…

Let $G$ be a semisimple Lie group, ${\frak g}$ its Lie algebra. For any symmetric space $M$ over $G$ we construct a new (deformed) multiplication in the space $A$ of smooth functions on $M$. This multiplication is invariant under the action…