Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space

Doctoral thesis (2020)

This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of ... [more ▼]

This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into holomorphic Riemannian space forms of constant curvature -1 (including SL(2,C) with a multiple of its Killing form): this leads to a Gauss-Codazzi theorem, it suggests an approach to holomorphic transitioning of immersions into pseudo-Riemannian space forms, a trick to construct holomorphic maps into the PSL(2,C)-character variety, and leads to a restatement of Bers theorem. The latter - a joint work with Andrea Seppi - consists in the study of immersions of n-manifolds inside the space of geodesics of the hyperbolic (n+1)-space. We give a characterization, in terms of the para-Kahler structure of this space of geodesics, of the Riemannian immersions which turn out to be Gauss maps of equivariant immersions into the hyperbolic space. [less ▲]

On immersions of surfaces into SL(2,C) and geometric consequences

E-print/Working paper (2020)

We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form ... [more ▼]

We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a similar version of the classic Gauss-Codazzi equations, and conversely that solutions of Gauss-Codazzi equations are immersion data of some equivariant map. This study has some interesting geometric consequences: 1) it provides a formalism to study immersions of surfaces into SL(2,C) and into the space of geodesics of H^3; 2) it generalizes the classical theory of immersions into non-zero curvature space forms, leading to a model for the transitioning of hypersurfaces among H^n, AdS^n, dS^n and S^n; 3) we prove that a holomorphic family of immersion data corresponds to a holomorphic family of immersions, providing an effective way to construct holomorphic maps into the SO(n,C)-character variety. In particular we will point out a simpler proof of the holomorphicity of the complex landslide; 4) we see how, under certain hypothesis, complex metrics on a surface (i.e. complex bilinear forms of its complexified tangent bundle) of constant curvature -1 correspond to pairs of projective surfaces with the same holonomy. Applying Bers Double Uniformization Theorem to this construction we prove a Uniformization Theorem for complex metrics on a surface. [less ▲]