arXiv subjects

Joseph Hoisington

Publications and source records attributed to Joseph Hoisington.

6 recordsLinked to original sources

arXiv2026

Area and antipodal distance in convex hypersurfaces

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere, proved by Berger in dimension $n=2$ and disproved by Croke in dimensions $n \geq 3$, is valid for convex hypersurfaces in all dimensions. We also establish a sharp lower bound for the mean width of a convex hypersurface.

math.DG↗

arXiv2024

Energy-minimizing mappings of real projective spaces

We give a sharp lower bound for the energy in homotopy classes of mappings from real projective space to Riemannian manifolds, together with an upper bound for its infimum. We characterize the maps which attain this lower bound for energy, and we explain how the infimum of the energy in a homotopy class of mappings of real projective n-space is determined by an associated class of mappings of the real projective plane.

math.DG↗

arXiv2023

Calibrations and Energy-Minimizing Mappings of Rank-1 Symmetric Spaces

We prove lower bounds for energy functionals of mappings from real, complex and quaternionic projective spaces to Riemannian manifolds. For real and complex projective spaces, these lower bounds are sharp, and we characterize the family of energy minimizing maps which arise in these results. We discuss the connections between these results and several theorems and questions in systolic geometry.

math.DG↗

arXiv2023

Energy-minimizing Mappings of Complex Projective Spaces

We show that in all homotopy classes of mappings from complex projective space to Riemannian manifolds, the infimum of the energy is proportional to the infimal area in the homotopy class of mappings of the 2-sphere which represents the induced homomorphism on the second homotopy group. We then establish a family of optimal lower bounds for a larger class of energy functionals for mappings from real and complex projective space to Riemannian manifolds and characterize the mappings which attain these lower bounds.

math.DG↗

arXiv2021

Hypersurfaces, Geodesics and Isoperimetric Inequalities in Cartan-Hadamard Manifolds

We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and we develop several facts about the spaces of geodesics in Cartan-Hadamards manifold that may be of independent interest.

math.DG↗

arXiv2017

Making matrices better: Geometry and topology of polar and singular value decomposition

Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal neighbor and nearest singular neighbor of a given matrix, both of which play central roles in matrix decompositions, and then against this visual backdrop examine the polar and singular value decompositions and some of their applications.

math.RA↗