arXiv subjects

Joseph Ansel Hoisington

Publications and source records attributed to Joseph Ansel Hoisington.

5 recordsLinked to original sources

arXiv2026

Mean curvature and closed geodesics in convex hypersurfaces

We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of Álvarez Paiva for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible.

math.DG↗

arXiv2026

Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in 1985. Our proof is based on an isometric embedding construction via holonomy, and uses Pogorelov's theory of surfaces with bounded extrinsic curvature. Along the way, we obtain a regularity result for convex hulls and a Schur-type comparison theorem for curves in Cartan-Hadamard manifolds.

math.DG↗

arXiv2021

Symmetry and Isoperimetry for Riemannian Surfaces

For a domain $Ω$ in a geodesically convex surface, we introduce a scattering energy $\mathcal{E}(Ω)$, which measures the asymmetry of $Ω$ by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving $\mathcal{E}(Ω)$ and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new proof of the sharp Sobolev inequality for Riemannian surfaces which is independent of the isoperimetric inequality.

math.DG↗

arXiv2021

Steiner's formula and a variational proof of the isoperimetric inequality

We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main difficulty in all of the proofs Steiner outlined for the plane isoperimetric inequality.

math.DG↗

arXiv2020

On the Total Curvature and Betti Numbers of Complex Projective Manifolds

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

math.DG↗