In: Geometriae Dedicata, 2008, vol. 134, no. 1, p. 177-196
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In: Mathematika, 2017, vol. 63, no. 1, p. 124–183
We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector...
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In: Manuscripta Mathematica, 2016, vol. 150, no. 3–4, p. 475–492
We prove that for every metric on the torus with curvature bounded from below by −1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is...
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In: Geometriae Dedicata, 2008, vol. 134, no. 1, p. 177-196
Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space ℍ³ and a group G of isometries of ℍ³ such that the induced metric on the quotient P/G is isometric to g. Moreover,...
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