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Jean-Marc Schlenker

Publications and source records attributed to Jean-Marc Schlenker.

81 records · Page 5Linked to original sources
arXiv2004

Hyperideal circle patterns

A ``hyperideal circle pattern'' in $S^2$ is a finite family of oriented circles, similar to the ``usual'' circle patterns but such that the closed disks bounded by the circles do not cover the whole sphere. Hyperideal circle patterns are directly related to hyperideal hyperbolic polyhedra, and also to circle packings. To each hyperideal circle pattern, one can associate an incidence graph and a set of intersection angles. We characterize the possible incidence graphs and intersection angles of hyperideal circle patterns in the sphere, the torus, and in higher genus surfaces. It is a consequence of a more general result, describing the hyperideal circle patterns in the boundaries of geometrically finite hyperbolic 3-manifolds (for the corresponding $\C P^1$-structures). This more general statement is obtained as a consequence of a theorem of Otal \cite{otal,bonahon-otal} on the pleating laminations of the convex cores of geometrically finite hyperbolic manifolds.

math.GT↗
arXiv2003

Hyperideal polyhedra in hyperbolic manifolds

Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings. The proof uses some elementary properties of the hyperbolic volume, in particular the Schläfli formula and the fact that the volume of (truncated) hyperideal simplices is a concave function of the dihedral angles.

math.GT↗
arXiv2003

Rhombic embeddings of planar graphs with faces of degree 4

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.

math-ph↗
arXiv2003

A rigidity criterion for non-convex polyhedra

Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid. More generally, let $P$ be a polyhedron bounding a domain which is the union of polytopes $C_1, ..., C_n$ with disjoint interiors, whose vertices are the vertices of $P$. Suppose that there exists an ellipsoid which contains no vertex of $P$ but intersects all the edges of the $C_i$. Then $P$ is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra.

math.DG↗
arXiv2002

Hyperbolic manifolds with polyhedral boundary

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$ has curvature $K>-1$, and each such metric on $\dr M$ is obtained for a unique choice of $g$. A dual statement is that, for each $g$ as above, the third fundamental form of $\dr M$ has curvature $K<1$, and its closed geodesics which are contractible in $M$ have length $L>2π$. Conversely, any such metric on $\dr M$ is obtained for a unique choice of $g$. We are interested here in the similar situation where $\partial M$ is not smooth, but rather looks locally like an ideal polyhedron in $H^3$. We can give a fairly complete answer to the question on the third fundamental form -- which in this case concerns the dihedral angles -- and some partial results about the induced metric. This has some by-products, like an affine piecewise flat structure on the Teichmueller space of a surface with some marked points, or an extension of the Koebe circle packing theorem to many 3-manifolds with boundary.

math.GT↗
arXiv2001

Hypersurfaces in $H^n$ and the space of its horospheres

A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric. This results extends in higher dimension, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced metric or the third fundamental form become fairly easy when one considers the horospherical metric. The results concerning the third fundamental form are obtained using a duality between $H^3$ and the de Sitter space $S^3_1$. In the same way, the results concerning the horospherical metric are proved through a duality between $H^n$ and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure.

math.DG↗
arXiv2000

On the Schlafli differential formula

he celebrated formula of Schlafli relates the variation of the dihedral angles of a smooth family of polyhedra in a space form and the variation of volume. We give a smooth analogue of this classical formula -- our result relates the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. The argument is direct, and the classical polyhedral result (as well as results for Lorenzian space forms) is an easy corollary. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary. We apply our results to extend the classical Euclidean inequalities of Aleksandrov to other 3-dimensional constant curvature spaces. We also obtain rigidity results for Ricci-flat manifolds with umbilic boundaries and existence results for foliations of Einstein manifolds by hypersurfaces.

math.DG↗
arXiv1999

Complete surfaces with negative extrinsic curvature

N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in $\R^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1 < \min(K_2, 0)$ such that $M$ contains no smooth, complete immersed surface with curvature below $K_1$. Optimal values of $K_1$ are determined. This results rests on a phenomenon of propagations for degenerations of solutions of hyperbolic Monge-Amp{è}re equations.

math.DG↗