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Ognjen Tošić

Publications and source records attributed to Ognjen Tošić.

7 recordsLinked to original sources
arXiv2025

On Hitchin's equations for cyclic G-Higgs bundles

We develop a Lie-theoretic perspective on Hitchin's equations for cyclic $G$-Higgs bundles, which we use to study analytic and geometric properties of harmonic maps. Among other things, we prove Dai-Li's conjecture on the monotonicity of the energy density in the case of Coxeter cyclic $G$-Higgs bundles, for all $G$, and Dai-Li's negative curvature conjecture for Coxeter cyclic $G$-Higgs bundles, for all $G$ except those of type $\mathrm{E}_7$ and $\mathrm{E}_8.$

math.DG↗
arXiv2025

Harmonic projections in negative curvature II: large convex sets

An important result in the theory of harmonic maps is due to Benoist--Hulin: given a quasi-isometry $f:X\to Y$ between pinched Hadamard manifolds, there exists a unique harmonic map at a finite distance from $f$. Here we show existence of harmonic maps under a weaker condition on $f$, that we call non-collapsing -- we require that the following two conditions hold uniformly in $x\in X$: (1) average distance from $f(x)$ to $f(y)$ for $y$ on the sphere of radius $R$ centered at $x$ grows linearly with $R$ (2) the pre-image under $f$ of small cones with apex $f(x)$ have low harmonic measures on spheres centered at $x$. Using these ideas, we also continue the previous work of the author on existence of harmonic maps that are at a finite distance from projections to certain convex sets. We show this existence in a pinched negative curvature setting, when the convex set is large enough. For hyperbolic spaces, this includes the convex hulls of open sets in the sphere at infinity with sufficiently regular boundary.

math.DG↗
arXiv2025

Constant energy families of harmonic maps

For a negatively curved manifold $M$ and a continuous map $ψ:Σ\to M$ from a closed surface $Σ$, we study complex submanifolds of Teichmüller space $\mathcal{S}\subset\mathcal{T}(Σ)$ such that the harmonic maps $\{h_X:X\to M\text{ for }X\in\mathcal{S}\}$ in the homotopy class of $ψ$ all have equal energy. When $M$ is real analytic with negative Hermitian sectional curvature, we show that for any such $\mathcal{S}$, there exists a closed Riemann surface $Y$, such that any $h_X$ for $X\in\mathcal{S}$ factors as a holomorphic map $ϕ_X:X\to Y$ followed by a fixed harmonic map $h:Y\to M$. This answers a question posed by both Toledo and Gromov. As a first application, we show a factorization result for harmonic maps from normal projective varieties to $M$. As a second application, we study homomorphisms from finite index subgroups of mapping class groups to $π_1(M)$.

math.DG↗
arXiv2024

Derivative of the Riemann-Hilbert map

Given a pair $(X,\nabla)$, consisting of a closed Riemann surface $X$ and a holomorphic connection $\nabla$ on the trivial principal bundle $X\times\mathrm{SL}_2(\mathbb{C})\to X$, the Riemann-Hilbert map sends $(X,\nabla)$ to its monodromy representation. We compute the derivative of this map, and provide a simple description of the locus where it is injective, recovering in the process several previously obtained results.

math.AG↗
arXiv2024

Non-strict plurisubharmonicity of energy on Teichmüller space

For an irreducible representation $ρ:π_1(Σ_g)\to\mathrm{GL}(n,\mathbb{C})$ there is an energy functional $\mathrm{E}_ρ:\mathcal{T}_g\to\mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\mathrm{GL}(n,\mathbb{C})/\mathrm{U}(n)$. It follows from a result of Toledo that $\mathrm{E}_ρ$ is plurisubharmonic, i.e. its Levi form is positive semi-definite. We study the kernel of this Levi form, and relate it to the $\mathbb{C}^*$ action on the moduli space of Higgs bundles. We also show that the points in $\mathcal{T}_g$ where strict plurisubharmonicity fails (i.e. this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,ρ)$, the map $\mathrm{E}_ρ$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.

math.CV↗
arXiv2023

Harmonic projections in negative curvature

In this paper we construct harmonic maps that are at a bounded distance from nearest-point retractions to convex sets, in negatively curved manifolds. Specifically, given a quasidisk $Q$ in hyperbolic space, we construct a harmonic map to the hyperbolic plane that corresponds to the nearest-point retraction to the convex hull of $Q$. If $M$ is a pinched Hadamard manifold so that its isometry group acts with cobounded orbits, and if $S$ is a set in the boundary at infinity of $M$, with the property that all elements of its orbit under the isometry group of $M$ have dimension less than $\frac{n-1}{2}$, we show that the nearest-point retraction to the convex hull of $S$ is a bounded distance away from some harmonic map.

math.DG↗