Proper quasi-homogeneous domains in flag manifolds and geometric structures

Zimmer, Andrew M.

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2635-2662.

On étudie dans cet article les domaines dans les variétés de drapeaux dont l’image dans une carte affine est bornée et dont l’action du groupe des automorphismes projectifs est co-compacte. Par contraste avec les nombreux exemples existant dans l’espace projectif réel, on démontre que de nombreuses variétés de drapeaux ne contiennent pas de tels domaines. On établit en outre que dans les cas où l’existence de tels domaines n’est pas exclue, ils sont soumis à une condition de convexité naturelle et possèdent une métrique invariante qui généralise la métrique de Hilbert. Une application de nos résultats fournit des restrictions sur l’application développante de certaines $(G,X)$-structures.

In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain $(G,X)$-structures.

Classification : 22F50, 53C24, 53A20
Keywords: Real projective structures, $(G,X)$-structure, Kobayashi metric, Carathéodory metric, Hilbert metric, projective automorphism group
Mot clés : structures projectives réelles, $(G,X)$-structure, métrique de Kobayashi, métrique de Carathéodory, métrique de Hilbert, automorphismes projectifs

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