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The Koebe-Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σg of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1,0 or −1 on the surface depending on whether g = 0,1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity.
In this paper, we consider projective structures on the surface where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing contains a neigborhood of the Koebe-Andreev-Thurston structure homeomorphic to ℝ6g−6. We furthemore show that if a circle packing consists of one circle, then the space is globally homeomorphic to ℝ6g−6 and that the circle packing is rigid.
We study in this paper sequences of solutions of elliptic PDE's with critical Sobolev growth on compact Riemannian manifolds. We prove some compactness results for such sequences which apply in particular to sequences of solutions of the Yamabe equation. We also underline the effect of the dimension and the geometry of the manifold on the blow-up behaviour of such sequences.
In this paper, we prove the existence of an isometric embedding near the origin in R3 of a two-dimensional metric with nonpositive Gaussian curvature. The Gaussian curvature can be allowed to be highly degenerate near the origin. Through the Gauss-Codazzi equations, the embedding problem is reduced to a 2 × 2 system of the first order derivaties and is solved via the method of Nash-Moser-Hörmander iterative scheme.