Metrics of Special Curvature with Symmetry

TL;DR

This paper explores special curvature metrics with symmetry, presenting new explicit extremal Kähler and Hermitian metrics, analyzing Einstein conditions, and developing methods for constructing and deforming such metrics on various manifolds.

Contribution

It introduces new extremal Kähler and Hermitian metrics, generalizes Kähler toric manifolds, and provides methods for constructing and deforming curvature metrics with symmetry.

Findings

Constructed explicit extremal Kähler metrics on fiberwise Kähler toric manifolds.

Developed a simplified Einstein condition system for 4-manifolds with $T^{2}$-symmetry.

Deformed the Fubini-Study metric to obtain new extremal Kähler metrics in dimension three.

Abstract

Various curvature conditions are studied on metrics admitting a symmetry group. We begin by examining a method of diagonalizing cohomogeneity-one Einstein manifolds and determine when this method can and cannot be used. Examples, including the well-known Stenzel metrics, are discussed. Next, we present a simplification of the Einstein condition on a compact four manifold with $T^{2}$-isometry to a system of second-order elliptic equations in two-variables with well-defined boundary conditions. We then study the Einstein and extremal Kahler conditions on Kahler toric manifolds. After constructing explicitly new extremal Kahler and constant scalar curvature metrics, we demonstrate how these metrics can be obtained by continuously deforming the Fubini-Study metric on complex projective space in dimension three. We also define a generalization of Kahler toric manifolds, which we call fiberwise Kahler toric manifolds, and construct new explicit extremal Kahler and constant scalar curvature metrics on both compact and non-compact manifolds in all even dimensions. We also calculate the Futaki invariant on manifolds of this type. After describing an Hermitian non-Kahler analogue to fiberwise Kahler toric geometry, we construct constant scalar curvature Hermitian metrics with $J$-invariant Riemannian tensor. In dimension four, we write down explicitly new constant scalar curvature Hermitian metrics with $J$-invariant Ricci tensor. Finally, we integrate the scalar curvature equation on a large class of cohomogeneity-one metrics.