Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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N. Rungi

Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space

created by rungi on 02 Dec 2023



Inserted: 2 dec 2023
Last Updated: 2 dec 2023

Year: 2023

ArXiv: 2309.09351 PDF


For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case $n=2$, we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in $\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal $\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated space is totally geodesic for any $n\ge 3$.

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