The covariance metric in the Blaschke locus. (English) Zbl 07827607
For a closed connected and oriented topological surface \(S\) of genus greater than one, let \(\mathcal{M}^B\) denote the space of Blaschke metrics on \(S\) and let \(\mathcal{M}_{-1}\) denote the space of smooth hyperbolic metrics on \(S\). A hyperbolic metric is a special case of a Blaschke metric on the surface. Considering the well-defined quotients of these spaces by the space \(\mathcal{D}_0\) of diffeomorphisms isotopic to the identity, the Teichmüller space \(\mathcal{T}(S)=\mathcal{M}_{-1}/\mathcal{D}_0\) can be viewed as a subset of the Blaschke locus \(\mathcal{M}^B/\mathcal{D}_0\). In this article, the Blaschke locus is the main object of study.
The first main result of the paper pertains to proving that the Blaschke locus is contractible and it is a smooth manifold of dimension \(16\mathcal{G}-17\) away from \(\mathcal{T}(S)\), where \(\mathcal{G}\) denotes the genus of \(S\). The proof uses techniques from affine differential geometry and Riemannian geometry, in particular, the regularity locus of the Blaschke locus is studied. The major tools implemented to obtain the results involve the construction of smooth charts of the Teichmüller space as demonstrated by A. J. Tromba [Teichmüller theory in Riemannian geometry: based on lecture notes by Jochen Denzler. Basel etc.: Birkhäuser Verlag (1992; Zbl 0785.53001)] and Wang’s equation from C. Wang [Lect. Notes Math. 1481, 271–280 (1991; Zbl 0743.53004)]. The latter refers to the second order partial differential equation \[ K(g)=-1+2 \vert q \vert ^2_g , \] where \(K(g)\) denotes the Gaussian curvature of a Blaschke metric \(g\), and \(\vert q \vert ^2_g =\frac{\vert q \vert ^2}{g^3}\) the pointwise \(g\)-norm of a holomorphic cubic differential \(q\). This equation provides a fundamental link between local slices for the bundle of holomorphic cubic differentials and local slices for the space of Blaschke metrics, thus allowing to produce smooth diffeomorphisms between these slices.
Yet, Wang’s equation can be viewed as a special case of Hitchin’s equation for the structure group \(\mathrm{PGL}(3, \mathbb{R})\), denoted here by \(\mathcal{H}_3(S)\). This allows one to view the Blaschke locus \(\mathcal{M}^B/\mathcal{D}_0\) as mapping class group equivariant homeomorphic to the quotient \(\mathcal{H}_3(S)/S^1\), for an \(S^1\)-action on \(\mathcal{H}_3(S)\). Thus, in the second part of the article, the authors treat the Riemannian geometry of the Blaschke locus with respect to the covariance metric by {C. Guillarmou} et al. [Ergodic Theory Dyn. Syst. 42, No. 3, 974–1022 (2022; Zbl 1493.37036)] called the pressure metric in [loc. cit.] Important results are obtained regarding the lengths of the covariant metric geodesics in the Blaschke locus.
The article is really well-written and demonstrates a wide collection of ideas and techniques from Riemannian and affine differential geometry.
The first main result of the paper pertains to proving that the Blaschke locus is contractible and it is a smooth manifold of dimension \(16\mathcal{G}-17\) away from \(\mathcal{T}(S)\), where \(\mathcal{G}\) denotes the genus of \(S\). The proof uses techniques from affine differential geometry and Riemannian geometry, in particular, the regularity locus of the Blaschke locus is studied. The major tools implemented to obtain the results involve the construction of smooth charts of the Teichmüller space as demonstrated by A. J. Tromba [Teichmüller theory in Riemannian geometry: based on lecture notes by Jochen Denzler. Basel etc.: Birkhäuser Verlag (1992; Zbl 0785.53001)] and Wang’s equation from C. Wang [Lect. Notes Math. 1481, 271–280 (1991; Zbl 0743.53004)]. The latter refers to the second order partial differential equation \[ K(g)=-1+2 \vert q \vert ^2_g , \] where \(K(g)\) denotes the Gaussian curvature of a Blaschke metric \(g\), and \(\vert q \vert ^2_g =\frac{\vert q \vert ^2}{g^3}\) the pointwise \(g\)-norm of a holomorphic cubic differential \(q\). This equation provides a fundamental link between local slices for the bundle of holomorphic cubic differentials and local slices for the space of Blaschke metrics, thus allowing to produce smooth diffeomorphisms between these slices.
Yet, Wang’s equation can be viewed as a special case of Hitchin’s equation for the structure group \(\mathrm{PGL}(3, \mathbb{R})\), denoted here by \(\mathcal{H}_3(S)\). This allows one to view the Blaschke locus \(\mathcal{M}^B/\mathcal{D}_0\) as mapping class group equivariant homeomorphic to the quotient \(\mathcal{H}_3(S)/S^1\), for an \(S^1\)-action on \(\mathcal{H}_3(S)\). Thus, in the second part of the article, the authors treat the Riemannian geometry of the Blaschke locus with respect to the covariance metric by {C. Guillarmou} et al. [Ergodic Theory Dyn. Syst. 42, No. 3, 974–1022 (2022; Zbl 1493.37036)] called the pressure metric in [loc. cit.] Important results are obtained regarding the lengths of the covariant metric geodesics in the Blaschke locus.
The article is really well-written and demonstrates a wide collection of ideas and techniques from Riemannian and affine differential geometry.
Reviewer: Georgios Kydonakis (Pátra)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C20 | Global Riemannian geometry, including pinching |
| 53A15 | Affine differential geometry |
| 58J05 | Elliptic equations on manifolds, general theory |
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