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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 $${\mathbb R}^3$$.

## Publications

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.