Solvability of Dirac type equations

TL;DR

This paper introduces a weighted $L^2$-method for Dirac equations, providing solvability conditions on Riemann surfaces and manifolds, with applications to holomorphic curves and eigenvalue estimates.

Contribution

It develops a new weighted $L^2$-approach for Dirac equations, offering solvability criteria, transversality conditions, and improved eigenvalue estimates in various geometric settings.

Findings

Sufficient curvature integral condition for Dirac equation solvability on Riemann surfaces.

Automatic transversality criterion for holomorphic curves in Kähler manifolds.

Enhanced eigenvalue estimates for Dirac operators with $Z_2$-grading.

Abstract

This paper develops a weighted $L^2$-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversality criteria for holomorphic curves in K\"ahler manifolds. On compact Riemannian manifolds, we give a new perspective on some well-known results about the first eigenvalue of the Dirac operator, and improve the estimates when the Dirac bundle has a $Z_2$-grading. On Riemannian manifolds with cylindrical ends, we obtain solvability in the $L^2$-space with suitable exponential weights while allowing mild negativity of the curvature.