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

Summary. The paper extends Patterson-Sullivan shadow estimates to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is global: the shadow center can be an arbitrary point in a Gromov model, including points deep in a cusp. 2

Main result. Theorem 1.4 gives a higher-rank relatively Morse analogue of the Stratmann-Velani global shadow lemma. For a theta-Morse subgroup and a Patterson-Sullivan measure associated to a linear form ψ with δψ(Γ)=1, the measure of a shadow centered in a horoball is comparable to explicit exponential factors plus a polynomial correction governed by the relevant peripheral subgroup. 5

Proof idea. The authors start from known orbit-shadow estimates for relatively Anosov subgroups, compare Gromov-model shadows with flag-variety shadows, and use peripheral counting asymptotics to derive the cusp correction. The same estimate yields uniform local estimates for Patterson-Sullivan measures and a comparison with Hausdorff measure for the associated visual quasi-metric. 5

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

Summary. The paper studies Gromov's exact-lift two-form method in scalar-curvature geometry. For a closed spin manifold carrying a homologically non-trivial closed two-form whose lift to the universal cover is exact, it proves a sharp scalar-curvature comparison with the bottom of the spectrum of the universal cover. 3

Main result. Theorem A says that, in even dimension at least 4, the infimum scalar curvature is bounded above by -4(2m)/(2m-1) times the bottom λ0 of the L2 spectrum on the universal cover. Equality forces the lifted metric to be Einstein; if λ0 > 0, the universal cover is real hyperbolic with the corresponding constant sectional curvature. 6

Proof idea. A twisted L2 index supplies harmonic spinors for small unitary twists of the Dirac operator. The scalar bound follows from the Lichnerowicz formula, refined Kato inequality, and Rayleigh characterization of λ0. For equality, the Kato defect is interpreted conformally; after a ground-state transform and recentering argument, a limiting parallel spinor gives Einstein rigidity. 6

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

Summary. The paper relates Lagrangian capacities of Liouville domains to Gutt-Hutchings capacities. It computes the Lagrangian capacity of convex and concave toric domains, settles the Cieliebak-Mohnke conjecture for ellipsoids, and proves boundary rigidity for extremal aspherical Lagrangians in ellipsoids. 4

Main result. Theorem 10 proves C^AL(X) <= inf_d C_d^GH(X)/d for Liouville domains with c1(X)=0. Corollary 12 computes C^CM(X)=C^AL(X)=diagonal(X) for convex or concave toric domains; for ellipsoids this becomes (1/a1 + ... + 1/an)^(-1). 7

Proof idea. The engine is a non-exact, S1-equivariant chain-level version of Viterbo functoriality. The authors construct an L-infinity structure on S1-equivariant string homology; for aspherical Lagrangians, degree constraints force holomorphic discs, and averaging their actions gives the C_d^GH(X)/d capacity bound. 7

Read Kim-Oh for Patterson-Sullivan theory in higher rank, Bei-Cecchini for scalar-curvature rigidity via exact-lift two-forms, and Faisal-Li for concrete symplectic-capacity computations and the chain-level string topology machinery behind them.