Kähler metric on the space of convex real projective structures on surface. (English) Zbl 1373.57045

Kim, Inkang; Zhang, Genkai

J. Differ. Geom. 106, No. 1, 127-137 (2017).

It has been conjectured for a long time that the Hitchin component admits a mapping class group invariant Kähler metric. The authors study this problem in the setting of surfaces. More precisely, they prove the Hitchin component of the character variety \(\chi (\pi_1(S), SL(3,\mathbb R))\) can be equipped with a mapping class group invariant Kähler metric where \(S\) is a closed surface of genus \(\geq 2\). Furthermore, let \(\mathcal T\) be the Teichmüller space of complex structures \(\Sigma_t\) on \(S\) with the holomorphic tangent space given by \(H^{(0,1)}(\Sigma_t,{\mathcal K}^{-1})\) at each \(t\in {\mathcal T}\), where \({\mathcal K}^{-1}\) is the tangent line bundle. If \(\mathcal T\) is equipped with the mapping class group invariant Weil-Petersson metric and the Hitchin component \(\chi_H(\pi_1(S), SL(3,\mathbb R))\) of the character variety is equipped with the mapping class group invariant Kähler metric as above then \(\mathcal T\) is a totally geodesic submanifold of \(\chi _H(\pi_1(S), SL(3,\mathbb R))\).

Reviewer: Neda Bokan (Beograd)

MSC:

57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57M60	Group actions on manifolds and cell complexes in low dimensions
53C55	Global differential geometry of Hermitian and Kählerian manifolds
57M50	General geometric structures on low-dimensional manifolds

Keywords:

character variety; Teichmüller space; Weil-Petersson metric; Griffith positivity; curvature positivity