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Minimizer of the Willmore energy with prescribed rectangular conformal class

The project concerns minimizers and critical points of the Willmore energy

$${\mathcal W}(f)=\int_MH^2dA$$,

i.e., the average value of the squared mean curvature $$H$$ of an immersion $$f\colon M\to{\mathbb R}^3$$ of a compact surface $$M$$. The Willmore conjecture that the minimum over tori is attained at the Clifford torus was recently proven by Marquez and Neves.

This project aims at determining all tori in 3-space minimizing the Willmore functional with prescribed rectangular conformal type by combining methods from Integrable Systems Theory and Global Analysis. Candidates are given by the real analytic family of constrained Willmore tori starting at the Clifford torus following the homogeneous tori, bifurcating at the first bifurcation point to the 2-lobed Delaunay tori and converging to the doubly covered sphere as the conformal type degenerates.

## 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.