Polarized 4-Manifolds, Extremal K\"ahler Metrics, and S-W Theory

TL;DR

This paper demonstrates that on certain 4-manifolds, Kaehler metrics with constant negative scalar curvature minimize scalar curvature energy, implying local symmetry on minimal ruled surfaces, using Seiberg-Witten theory.

Contribution

It establishes a minimization property of Kaehler metrics with negative scalar curvature on 4-manifolds via Seiberg-Witten invariants, linking geometric analysis and topology.

Findings

Kaehler metrics of constant negative scalar curvature minimize scalar curvature energy

Such metrics on minimal ruled surfaces are locally symmetric

Seiberg-Witten theory is used to prove these minimization properties

Abstract

Using Seiberg-Witten theory, it is shown that any Kaehler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L^2-norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H^2(M)=(H^+) + (H^-). This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric.