Existence, regularity and uniqueness results of geometric variational problems

The regularity of area minimising currents generated remarkable mathematics in the last century. One of the cornerstones is the proof of the partial regularity theorem for area minimising currents in higher codimension without boundary by Almgren. The question of boundary regularity is still open in higher codimensions. This project investigates possible new approaches, for example the regularity of so-called \(Q-\frac{1}{2}\) Dir-minimisers.

Another problem is about harmonic maps and their homotopy classes and, in particular, extending the approach of Luckhaus to the \(H_\omega^{1,p}\)setting.

A third problem concerns extending existence and classification results for orientable Willmore surfaces to the non-orientable case and, in particular, to prove the existence and smoothness of a Willmore minimising Klein bottle in \({\mathbb R}^3\).


We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of Higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball, and we compute its dimension explicitly. For example, the Hitchin component of the right-angled hyperbolic \(\ell\)-polygon reflection group into \(\mathrm{PGL}(2m,\mathbb{R})\), resp. \(\mathrm{PGL}(2m+1,\mathbb{R})\), is homeomorphic to an open ball of dimension \((\ell-4)m^2+1\), resp. \((\ell-4)(m^2+m)\). We also give applications to the study of the pressure metric and the deformation theory of real projective structures on 3-manifolds.


Related project(s):
1Hitchin components for orbifolds

Geometric structures on manifolds became popular when Thurston used them in his work on the Geometrization Conjecture. They were studied by many people and they play an important role in Higher Teichmüller Theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Thesis that this technique can be used to construct geometric structures in interesting cases. Here we will survey some recent results in this direction, which are joint work with Qiongling Li.


Related project(s):
1Hitchin components for orbifolds

For d = 4, 5, 6, 7, 8, we exhibit examples of \(\mathrm{AdS}^{d,1}\) strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space \(\mathbb{H}^d\), nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot-Mérigot [BM12]. We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] and find examples of Coxeter groups W such that the space of strictly GHC-regular representations of W into \(\mathrm{PO}_{d,2}(\mathbb{R})\) up to conjugation is disconnected.


Related project(s):
1Hitchin components for orbifolds

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Team Members

Dr. Jonas Hirsch
Project leader
Universität Leipzig

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