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Georgios Daskalopoulos

Publications and source records attributed to Georgios Daskalopoulos.

17 recordsLinked to original sources
arXiv2025

Harmonic maps in singular geometry and rigidity

This survey reviews results on harmonic maps into spaces of non-positive curvature, with a focus on targets that lack smooth structure. More precisely, we consider targets that are complete metric spaces with non-positive curvature in the sense of Alexandrov, commonly referred to as NPC (non-positively curved) or CAT(0) spaces. We discuss applications of harmonic maps to rigidity phenomena, including generalizations of Margulis superrigidity and the holomorphic rigidity of Teichm\"uller space. Our approach relies heavily on the regularity theory of harmonic maps to non-smooth targets, enabling differential-geometric techniques to be employed in the absence of any smooth structure on the target.

math.DG↗
arXiv2024

Best Lipschitz maps and Earthquakes

This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces (infinity harmonic maps), which induce dual Lie algebra valued transverse measures with support on Thurston's canonical lamination. The present paper examines these Lie algebra valued measures in greater detail. For any measured lamination we are led to define a Lie algebra valued measure and conversely every Lie algebra valued transverse measure arrises from this process. Furthermore, we show that such measures are infinitesimal earthquakes. This construction provides a natural correspondence between best Lipschitz maps and earthquakes.

math.DG↗
arXiv2023

Notes on Harmonic Maps

This is set of notes prepared for the Summer School on non-Abelian Hodge theory in Abbaye de Saint-Jacut de la Mer June, 6-19, 2022. We cover the following topics: Lecture 1. Harmonic Maps Between Riemannian Manifolds Lecture 2. Existence and Regularity Lecture 3. Pluriharmonic Maps and the Siu-Sampson Formula Lecture 4. Donaldson Corlette Theorem

math.DG↗
arXiv2022

Pluriharmonic maps into buildings and symmetric differentials

Given a complex smooth quasi-projective variety $X$, a semisimple algebraic group $G$ defined over some non-archimedean local field $K$ and a Zariski dense representation $\varrho:\pi_1(X)\to G(K)$, we construct a $\varrho$-equivariant (pluri-)harmonic map from the universal cover of $X$ into the Bruhat-Tits building $\Delta(G)$ of $G$, with some suitable asymptotic behavior. This theorem generalizes the previous work by Gromov-Schoen to the quasi-projective setting. As an application, we prove that $X$ has nonzero global logarithmic symmetric differentials if there exists a linear representation $\pi_1(X)\to {\rm GL}_N(\mathbb{K})$ with infinite image, where $ \mathbb{K}$ is any field. This theorem generalizes the previous work by Brunebarbe, Klingler and Totaro to the quasi-projective setting.

math.AG↗
arXiv2022

Analytic properties of Stretch maps and geodesic laminations

In a 1998 preprint, Bill Thurston outlined a Teichmuller theory for hyperbolic surfaces based on maps between surfaces which minimize the Lipschitz constant (minimum stretch or best Lipschitz maps). In this paper we continue the analytic investigation which we began in our previous paper. In the spirit of the construction of infinity-harmonic functions, we produce best Lipschitz maps u as limits p goes to infinity of minimizers of p-Schatten integrals (p-Schatten harmonic maps) in a fixed homotopy class between hyperbolic surfaces. We address existence and regularity of p-Schatten harmonic maps with the latter, due to higher degeneracies, being significantly harder than for ordinary p- harmonic maps. Moreover, we construct Lie algebra valued dual functions which minimize a dual q-Schatten integral and limit as q goes to 1 to a locally defined, Lie algebra valued function v of bounded variation. One of the main results of the paper is the surprising fact that the support of the measure dv (the derivative of v) lies on the canonical geodesic lamination constructed by Thurston and further studied by Gueritaud-Kassel. In the sequel paper we will show how these Lie algebra valued measures induce a transverse measure on the canonical lamination and relate to other aspects of Thurston theory.

math.DG↗
arXiv2022

Transverse Measures and Best Lipschitz and Least Gradient Maps

We exhibit the duality between best Lipschitz (infinity harmonic) maps and least gradient maps in the case of maps from surfaces to the circle. We show that given a homotopy class of a map from a surface to the circle the infinity harmonic map defines a geodesic lamination on the surface and the dual least gradient map defines a transverse measure on the lamination. This is the initial step towards an analytic approach to Thurston's work on best Lipschitz maps between hyperbolic surfaces and Thurston's asymmetric metric on Teichmueller space.

math.DG↗
arXiv2021

Infinite energy maps and rigidity

We extend Siu's and Sampson's celebrated rigidity results to non-compact domains. More precisely, let $M$ be a smooth quasi-projective variety with universal cover $\tilde M$ and let $\tilde X$ be a symmetric space of non-compact type, a locally finite Euclidean building or the Weil-Petersson completion of the Teichmüller space of a surface of genus $g$ and $p$ punctures with $3g-3+p>0$. Under suitable assumptions on a homomorphism $ρ: π_1(M) \rightarrow \mathsf{Isom}(\tilde X)$, we show that there exists a $ρ$-equivariant pluriharmonic map $\tilde u: \tilde M \rightarrow \tilde X$ of possibly infinite energy. In the case when the target is Kähler and $\mathsf{rank}(d \tilde u) \geq 3$ at some point, $\tilde u$ is holomorphic or conjugate holomorphic. This builds on previous important work by Jost-Zuo and Mochizuki. We also extend these results to the case when the target is a Riemannian manifold with sectional curvature bounded from above by a negative constant.

math.DG↗
arXiv2020

Rigidity of Teichmuller Space

We prove the holomorphic rigidity conjecture of Teichmüller space which loosely speaking states that the action of the mapping class group uniquely determines the Teichmüller space as a complex manifold. The method of proof is through harmonic maps. We prove that the singular set of a harmonic map from a smooth $n$-dimensional Riemannian domain to the Weil-Petersson completion $\overline{\mathcal T}$ of Teichmüller space has Hausdorff dimension at most $n-2$, and moreover, $u$ has certain decay near the singular set. Combining this with the earlier work of Schumacher, Siu and Jost-Yau, we provide a proof of the holomorphic rigidity of Teichmüller space. In addition, our results provide as a byproduct a harmonic maps proof of both the high rank and the rank one superrigidity of the mapping class group proved via other methods by Farb-Masur and Yeung.

math.DG↗
arXiv2009

Morse Theory and Hyperkahler Kirwan Surjectivity for Higgs Bundles

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott's original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperkähler Kirwan map is surjective for the non-fixed determinant case.

math.SG↗
arXiv2002

Classification of Weil-Petersson Isometries

This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the Thurston classification. The second result concerns the asymptotic behavior of these geodesics. It is shown that geodesics that are equivariant with respect to independent pseudo-Anosov's diverge. It follows that subgroups of the mapping class group which contain independent pseudo-Anosov's act in a reductive manner with respect to the Weil-Petersson geometry. This implies an existence theorem for equivariant harmonic maps to the metric completion.

math.DG↗
arXiv1998

The Yang-Mills flow near the boundary of Teichmueller space

We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of SU(2) representations of a once punctured surface. This foliation universalizes over Teichmüller space and is equivariant with respect to the action of the mapping class group. It is shown how to extend the foliation as a singular foliation over the Strebel boundary of Teichmüller space, and continuity of this extension is the main result of the paper.

math.DG↗
arXiv1998

Character varieties and harmonic maps to R-trees

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\mathbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\mathbb R}$-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.

math.DG↗
arXiv1995

On the Brill-Noether Problem for Vector Bundles

On an arbitrary compact Riemann surface, necessary and sufficient conditions are found for the existence of semistable vector bundles with slope between zero and one and a prescribed number of linearly independent holomorphic sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional.

alg-geom↗
arXiv1993

Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective scheme and is dominated by the algebraic compactification arising as a Grothendieck Quot scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as Gromov invariants) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one.

alg-geom↗