arXiv subjects

Ivo Slegers

Publications and source records attributed to Ivo Slegers.

4 recordsLinked to original sources

arXiv2021

Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations

Let $Σ$ be a closed surface of genus least two and $ρ\colon π_1(Σ) \to G$ a Hitchin representation into $G=\text{PSL}(n,\mathbb{R})$, $\text{PSp}(2n,\mathbb{R})$, $\text{PSO}(n,n+1)$ or $\text{G}_2$. We consider the energy functional $E$ on the Teichmüller space of $Σ$ which assigns to each point in $\mathcal{T}(Σ)$ the energy of the associated $ρ$-equivariant harmonic map. The main result of this paper is that $E$ is strictly plurisubharmonic. As a corollary we obtain an upper bound of $3 \cdot \text{genus}(Σ) -3$ on the index of any critical point of the energy functional.

math.DG↗

arXiv2021

Exponential convergence rate of the harmonic heat flow

We consider the harmonic heat flow for maps from a compact Riemannian manifold into a Riemannian manifold that is complete and of non-positive curvature. We prove that if the harmonic heat flow converges to a limiting harmonic map that is a non-degenerate critical point of the energy functional, then the rate of convergence is exponential (in the $L^2$ norm).

math.DG↗

arXiv2021

The energy spectrum of metrics on surfaces

Let $(N,ρ)$ be a Riemannian manifold, $S$ a surface of genus at least two and let $f\colon S \to N$ be a continuous map. We consider the energy spectrum of $(N,ρ)$ (and $f$) which assigns to each point $[J]\in \mathcal{T}(S)$ in the Teichmüller space of $S$ the infimum of the Dirichlet energies of all maps $(S,J)\to (N,ρ)$ homotopic to $f$. We study the relation between the energy spectrum and the simple length spectrum. Our main result is that if $N=S$, $f=id$ and $ρ$ is a metric of non-positive curvature, then the energy spectrum determines the simple length spectrum. Furthermore, we prove that the converse does not hold by exhibiting two metrics on $S$ with equal simple length spectrum but different energy spectrum. As corollaries to our results we obtain that the set of hyperbolic metrics and the set of singular flat metrics induced by quadratic differentials satisfy energy spectrum rigidity, i.e. a metric in these sets is determined, up to isotopy, by its energy spectrum. We prove that analogous statements also hold true for Kleinian surface groups.

math.DG↗

arXiv2020

Equivariant harmonic maps depend real analytically on the representation

We prove that when assuming suitable non-degeneracy conditions equivariant harmonic maps into symmetric spaces of non-compact type depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in \cite{EellsLemaire} can be applied.

math.DG↗