arXiv subjects

Samuel Lockman

Publications and source records attributed to Samuel Lockman.

4 recordsLinked to original sources

arXiv2026

Rigidity for spin fill-ins with scalar curvature bounded from below

We establish the rigidity statement in the equality case of the hyperspherical-radius inequality of Brendle, Tsiamis, and Wang for compact spin fill-ins with scalar curvature bounded below. More precisely, let $(M^{n\geq 3},g)$ be a compact, connected Riemannian spin manifold having a connected boundary $\Sigma$ and scalar curvature satisfying $\mathrm{scal}_g\geq -n(n-1)$. We prove that equality in the upper bound \[ \inf_{\Sigma}H\leq (n-1)\sqrt{1+\operatorname{Rad}(\Sigma)^{-2}} \] given by Brendle, Tsiamis, and Wang holds if and only if $(M,g)$ is isometric to a geodesic ball in hyperbolic space.

math.DG↗

arXiv2026

Scalar curvature rigidity for products of convex hypersurfaces

Let $N = N_{1} \times \dotsm \times N_{k}$, where each $N_{i} \subset \mathbb{R}^{n_i+1}$ is a closed strictly convex hypersurface. Let $M$ be a Riemannian spin manifold of dimension $n = \dim(N)$, and let $f \colon M \to N$ be an area non-increasing smooth map of non-zero degree. We show that $\mathrm{scal}_M \geq \mathrm{scal}_N \circ f$ implies $\mathrm{scal}_M = \mathrm{scal}_N \circ f$. Moreover, if $n \geq 3$ and $N$ has no circle factors, then every such map is a Riemannian isometry. In the presence of circle factors, we obtain the corresponding optimal splitting theorem for $M$ and $f$. Our results are based on an approach to the index-theoretic part of Llarull's scalar curvature rigidity theorem via Clifford-linear family index theory, which works independently of the parity of the dimension and extends naturally to products. This includes a proof of the Geroch conjecture for spin manifolds as the edge case with only circle factors.

math.DG↗

arXiv2026

A non-trivial index difference on surfaces of genus at least $3$

For every closed surface of genus at least $3$, equipped with any bounding spin structure, we show that the index difference, viewed as a map from the fundamental group of the space of Dirac-invertible Riemannian metrics to $\KO^{-4}(*)$, is non-trivial. If the genus is at least $5$, we also show that the aforementioned index difference is surjective. For products of two closed surfaces of genus at least $3$, equipped with any spin structure, we prove that the corresponding space of Dirac-invertible Riemannian metrics is not contractible. We discuss the relationship of this result to the existence of metrics with harmonic spinors in dimension~$4$.

math.DG↗

arXiv2025

Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor

We classify Riemannian $\text{spin}^c$ manifolds carrying a type I imaginary generalized Killing spinor, by explicitly constructing a parallel spinor on each leaf of the canonical foliation given by the Dirac current. We also provide a class of Riemannian $\text{spin}^c$ manifolds carrying a type II imaginary generalized Killing spinor, by considering spacelike hypersurfaces of Lorentzian $\text{spin}^c$ manifolds. We carry out much of the work in the setting of semi-Riemannian $\text{spin}^c$-manifolds carrying generalized Killing spinors, allowing us to draw conclusions in this setting as well. In this context, the Dirac current is not always a closed vector field. We circumvent this in even dimensions, by considering a modified Dirac current, which is closed in the cases when the original Dirac current is not. On the path to these results, we also study semi-Riemannian manifolds carrying closed and conformal vector fields.

math.DG↗