Drawing bananas!

My Ph.D. thesis is about twisted harmonic maps, and more specifically, deformations of such. If $M$ is a Riemannian manifold and $\Gamma = \pi_1(M, x_0)$ its fundamental group, letting $\rho \colon \Gamma \to G$ be a representation to a semisimple (or, more generally, reductive) group $G$, a twisted harmonic map is a map $$f \colon \tilde{M} \to N = G/K$$ where $\tilde{M}$ is the universal cover of $M$, and we suppose that $f(\gamma \tilde x) = \rho(\gamma) f(\tilde{x})$ and that $f$ is harmonic, that is, it satisfies $\text{trace} \nabla \text{d} f = 0$. Here we see $\text{d}f$ as a $f^*TN$-valued 1-form and $\nabla$ is the pull-back of the Levi-Civita connection of $N$. Equivalently, $f$ being harmonic means that it is a critical point for the energy functional $E(f) = \frac{1}{2} \int_M \| \text{d} f\|^2 \text{d Vol}$ among all smooth maps with the same kind of equivariance (the integral is taken over any fundamental region for the action of $\Gamma$ on $\tilde M$). The symmetric space $N$ being of non-positive curvature, such maps are automatically energy-minimizing. Corlette's theorem [1]K. Corlette Flat $G$-bundles with canonical metrics. J. Differential Geom. 28(3):361--382. implies that these maps exist if and only if the representation $\rho$ is reductive, i.e. a direct sum of irredicuble representations.

We are interested here in a specific example: The case where $M = S^1$. Then, $\Gamma = \mathbb{Z}$ and $\tilde M = \mathbb{R}$, and harmonic maps are simply geodesics. A representation is tantamount to a single element $g \in G$, and a $g$-equivariant geodesic is then a geodesic $f \colon \mathbb R \to N$ such that $f(x + 1) = g \cdot f(x)$. A "reductive representation" now means simply a semisimple (i.e. diagonalizable) element $g$. One sees easily that the energy of $g$, which is defined as the infimum of the energy of $g$-equivariant maps (hence, for semisimple $g$, the energy of any $g$-equivariant geodesic) is the square of the translation length of the isometry $g$: $$E(g) = \inf_{y \in N} \text{dist}\big( y, g\cdot y\big)^2.$$ In one direction, it is well known that the length of any curve joining $y$ to $g \cdot y$ is at least the distance between $y$ and $g\cdot y$; in the other direction, we can join these two points by the unique geodesic arc $f \colon [0,1] \to N$ and extend the definition on $\mathbb R$ by $f(x) = g^n f(x - n)$, where $n = \lfloor x \rfloor$. The obtained curve is not necessarily smooth (in fact, it is smooth if and only if it is the unique geodesic); however, we can approximate it arbitrarily well by smooth maps, which then must realize the infimum. In this way we obtain, in this very special case $M = S^1$, that Corlette's theorem reduces to the well known fact that $g$ being semisimple is equivalent to the translation length attaining a minimum.

When $G = \text{SL}(2, \mathbb R)$, we know the formula for the distance between two points of $N$, which is the hyperbolic plane. Using this fact, we give below a utility which, given a matrix $$A = \begin{pmatrix} a & b\\c & d\end{pmatrix}, \quad \det(A) > 0$$ computes for each point $z$ in the upper half plane the distance between $z$ and $A\cdot z = \frac{az+b}{cz+d}$. In the plot, we set as pure black the minimum value taken by the function; ideally, this would print in black only the points in the support of the geodesic, which are exactly those where the energy reaches the absolute minimum (higher sharpness values, up to 100, delimit the geodesic better, while lower values, down to 5, tend to color the whole of the figure). One can try different values for the coefficients of $A$; if the matrix is non-diagonalizable (e.g. $a=b=d=1$, $c=0$) one sees that the minimum locus "escapes to infinity". Giving small positive values to $c$, one can even see the geodesic getting larger and "spreading" to a horizontal line.

The main subject of my thesis involves computing the deformation of the twisted harmonic map with respect to a deformation of the representation $\rho$; this result is then applied to compute the variation of the energy functional on the moduli space. In the case of $M = S^1$, the representation space $\text{Hom}(\Gamma, G)$ reduces to $G$, and a deformation of a representation $g$ is an element $\xi \in \mathfrak{g}$, the Lie algebra of $G$, defined by $\xi = \frac{\partial g_t}{\partial t}\big|_{t=0} g^{-1}$. Then, Hodge theory grants the existence of a harmonic 1-form representing $\xi$; in this case, this can be arranged to give a map $\eta \colon \mathbb R \to \mathfrak g$ such that: $$\eta(x + 1) = \text{Ad}_g \eta(x); \qquad \int_{x_0}^{x_0+1} \eta(x) \text{d} x = \xi + \text{coboundary};\qquad \eta'(x) = 2 \big[f'(x), \eta(x)\big],$$ where $f'(x) \in T_{f(x)}N$ is seen as an element of $\mathfrak g$ via the canonical embedding $TN \hookrightarrow N \times \mathfrak g$, i.e. the right inverse to the map $(n,\xi)\mapsto \frac{\partial}{\partial t} \big(\exp(t\xi)\cdot n\big)$. In terms of this $\eta$ and of the Killing form on $\mathfrak g$, the variation of the energy (that is, of the square of the translation length) along $\xi$ is given by: $$\frac{\partial E}{\partial t}\Big|_{t=0} = \int_0^1 \text{Kill}\Big(\eta(x), f'(x)\Big) \text{d} x.$$