math.GT daily digest: 3 new submissions for 19 June 2026

The Friday, 19 June 2026 arXiv math.GT listing has 3 eligible new papers: 1 primary math.GT submission and 2 cross-lists carrying math.GT. The 4 replacement submissions on the page are excluded because this channel tracks new submissions and cross-lists only. 1

Authors: Dongryul M. Kim and Hee Oh arXiv: 2606.19779 Subjects: math.GT, math.DS, math.GR 2

Abstract. Patterson-Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. In this paper, we prove a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric. 2

Authors: Francesco Bei and Simone Cecchini arXiv: 2606.19567 Subjects: math.DG, math.GT 3

Abstract. We study Gromov's exact-lift two-form method in scalar-curvature geometry. For a closed Riemannian spin manifold carrying a homologically non-trivial closed two-form whose lift to the universal cover is exact, we prove the sharp hyperbolic scalar-curvature comparison with the bottom of the spectrum of the universal Riemannian covering. The two-form enters through Gromov's twisted -index, which produces harmonic spinors for a family of small unitary twists. We analyze the equality case by interpreting the refined Kato equality defect conformally and use the harmonic spinors to construct a parallel spinor with respect to a suitable conformally related metric. This yields that the original metric is Einstein. In the positive-spectrum case, this method implies that the universal cover is real hyperbolic. 3

Authors: Shah Faisal and Yin Li arXiv: 2606.20051 Subjects: math.SG, math.GT 4

Abstract. We derive upper bounds for the Lagrangian capacities of Liouville domains with finite Gutt--Hutchings capacities and show that the Lagrangian capacity of a convex or concave toric domain of arbitrary dimension equals its diagonal. In particular, this completely settles the conjecture of Cieliebak-Mohnke on the Lagrangian capacity of ellipsoids. Our proof is based on an -equivariant variant of the techniques of Fukaya and Irie, and does not use holomorphic curves with local tangency constraints, which would inevitably cause transversality issues. Moreover, we show that any extremal Lagrangian torus in an $n$-dimensional ellipsoid must lie on the boundary. Applications of our results and techniques include new upper bounds on the Lagrangian width for aspherical Lagrangians in Liouville manifolds and the first computations of the Lagrangian capacities for many non-subcritical Weinstein domains in dimensions 4 and 6. 4