A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature

2025/10/15 by Zehao Sha, Sha, Zehao
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Compact Riemann surface #Compact space #Curvature #Differential geometry #Fibered knot #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic function #Mean curvature #Scalar curvature

paper · pdf · doi:10.48550/arxiv.2510.13353

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/08/09

Abstract

In this note, we prove a 2-systolic inequality on compact positive scalar curvature Kähler surfaces admitting a nonconstant holomorphic map to a positive-genus compact Riemann surface. According to the classification of positive scalar curvature Kähler surfaces, any such surface must be a ruled surface fibred over a complex curve with positive genus.

Citations

• Systolic Inequality and Scalar Curvature
• The sign of scalar curvature on Kähler blowups
• A topological gap theorem for the π₂-systole of positive scalar curvature 3-manifolds
• On the 2-systole of stretched enough positive scalar curvature metrics on S 2 x S 2
• Scalar curvature and harmonic maps to S¹
• Rigidity of area-minimizing two-spheres in three-manifolds
• Area-minimizing projective planes in three-manifolds
• On the Scalar Curvature of Complex Surfaces
• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature

Related

• Systolic Inequality and Scalar Curvature
• Stability of systolic inequalities for the Möbius strip and Klein bottle
• Homological n-systole in (n+1)-manifolds and bi-Ricci curvature
• A remark on Kähler metrics of constant scalar curvature on ruled complex surfaces
• Dyck's surfaces, systoles, and capacities