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Analysis on spaces with fibred cusps

The main object of this project are manifolds equipped with a certain class of complete Riemannian metrics, which we call fibred cusp metrics, and generalizations and variants of them. 

A fibred cusp metric on a compact manifold with boundary \(M\), and associated to a fibration \(\phi\colon\partial M\to B\) is a Riemannian metric on the interior of \(M\)which near \(\partial M\) takes the form  

\(g=\frac{dx^2}{x^2}+\phi^*g_B+x^{2\alpha}g_F\)

in terms of a trivialization \(\left[0,\epsilon\right)\times\partial M\) of a neighborhood of \(\partial M\), where \(x\) is the coordinate on \(\left[0,\epsilon\right)\),  \(g_B\) is a Riemannian metric on \(B\) and \(g_F\) is a 'fibre metric' for \(\phi\) (i.e. a symmetric 2-tensor on \(\partial M\) which restricts to a metric on each fibre of \(\phi\)). Generalizations involve a tower of fibrations \(\partial M\to B_{m-1}\to\ldots\to B_0\) with different orders of degeneration for each level of fibre, or a manifold with corners \(M\), where each boundary hypersurface carries fibrations. Variants include conformally equivalent metrics such as \(x^{-2\alpha}g\) (so-called fibred boundary metric).

Such metrics occur on locally symmetric spaces and various moduli spaces, for example. The boundary \(\partial M\), i.e. \(x = 0\), corresponds to 'infinity'.

The goals of this project are to develop analytic tools for analyzing the natural geometric differential operators associated to such metrics and to apply these tools to problems in global analysis and spectral theory. In particular, pseudodifferential calculi and heat calculi adapted to such metrics will be constructed.


Publications

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

Malte Behr
Doctoral student
Carl-von-Ossietzky-Universität Oldenburg
malte.behr(at)uni-oldenburg.de

Prof. Dr. Daniel Grieser
Project leader
Carl-von-Ossietzky-Universität Oldenburg
daniel.grieser(at)uni-oldenburg.de

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