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Peter A. Storm

Publications and source records attributed to Peter A. Storm.

12 recordsLinked to original sources
arXiv2010

Lipschitz minimality of Hopf fibrations and Hopf vector fields

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.

math.GT↗
arXiv2010

Moduli spaces of hyperbolic 3-manifolds and dynamics on character varieties

The space AH(M) of marked hyperbolic 3-manifold homotopy equivalent to a compact 3-manifold with boundary M sits inside the PSL_2(C)-character variety X(M) of π_1(M). We study the dynamics of the action of Out(π_1(M)) on both AH(M) and X(M). The nature of the dynamics reflects the topology of M. The quotient AI(M)=AH(M)/Out(π_1(M)) may naturally be thought of as the moduli space of unmarked hyperbolic 3-manifolds homotopy equivalent to M and its topology reflects the dynamics of the action.

math.GT↗
arXiv2009

Rigidity of minimal volume Alexandrov spaces

Let Z be an Alexandrov space with curvature bounded below by -1 such that Z is homotopy equivalent to a real hyperbolic manifold M. It is known that the volume of Z is not smaller than the volume of M. If the volumes are equal, this short paper proves that the homotopy equivalence is homotopic to an isometric homeomorphism. The main analytic tool is a theorem of Reshetnyak about quasiregular maps.

math.GT↗
arXiv2009

Hyperbolic convex cores and simplicial volume

This paper investigates the relationship between the topology of hyperbolizable 3-manifolds M with incompressible boundary and the volume of hyperbolic convex cores homotopy equivalent to M. Specifically, it proves a conjecture of Bonahon stating that the volume of a convex core is at least half the simplicial volume of the doubled manifold DM, and this inequality is sharp. This paper proves that the inequality is in fact sharp in every pleating variety of AH(M).

math.GT↗
arXiv2009

The barycenter method on singular spaces

Bonahon conjectured that compact convex cores with totally geodesic boundary uniquely minimize volume over all hyperbolic 3-manifolds in the same homotopy class. This paper proves Bonahon's conjecture. The proofs extend the techniques of Besson-Courtois-Gallot.

math.GT↗
arXiv2009

Minimal volume Alexandrov spaces

Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by -1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, closed hyperbolic manifolds minimize volume over all hyperbolic cone manifolds in the same bilipschitz class with cone angles not greater than 2pi. The proof uses techniques developed by Besson-Courtois-Gallot. In 3 dimensions, this result provides a partial solution to a conjecture in Kleinian groups concerning acylindrical manifolds.

math.GT↗
arXiv2008

From the hyperbolic 24-cell to the cuboctahedron

We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry group of hyperbolic 4-space. It also leads to finite covolume Coxeter groups which are the homomorphic image of the group of reflections in the hyperbolic 24-cell. The examples are constructed very explicitly, both from an algebraic and a geometric point of view. The method used can be viewed as a 4-dimensional, but infinite volume, analog of 3-dimensional hyperbolic Dehn filling.

math.GT↗
arXiv2005

Lower bounds on volumes of hyperbolic Haken 3-manifolds

We prove a volume inequality for 3-manifolds having C^0 metrics "bent" along a hypersurface, and satisfying certain curvature pinching conditions. The result makes use of Perelman's work on Ricci flow and geometrization of closed 3-manifolds. Corollaries include a new proof of a conjecture of Bonahon about volumes of convex cores of Kleinian groups, improved volume estimates for certain Haken hyperbolic 3-manifolds, and a lower bound on the minimal volume orientable hyperbolic 3-manifold. An appendix by Dunfield compares estimates of volumes of hyperbolic 3-manifolds drilled along a closed embedded geodesic with experimental data.

math.DG↗