arXiv subjects

Michael G. Dabkowski

Publications and source records attributed to Michael G. Dabkowski.

4 recordsLinked to original sources

arXiv2016

On the proportionality of Chern and Riemannian scalar curvatures

On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to general Hermitian manifolds. This idea has been the subject of much study from the cohomological side, however, the focus here is to address such a question from the perspective of curvature relationships. In particular, on compact manifolds the Kähler condition is characterized by the relationship that the Chern scalar curvature is equal to half the Riemannian scalar curvature. What we study here is the existence, or lack thereof, of non-Kahler Hermitian metrics for which a more general proportionality relationship between these scalar curvatures holds.

math.DG↗

arXiv2016

The lowest eigenvalue of Schrödinger operators on compact manifolds

The lowest eigenvalue of the Schrödinger operator $-Δ+\mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.

math.DG↗

arXiv2015

On Kähler conformal compactifications of $U(n)$-invariant ALE spaces

We prove that a certain class of ALE spaces always has a Kahler conformal compactification, and moreover provide explicit formulas for the conformal factor and the Kahler potential of said compactification. We then apply this to give a new and simple construction of the canonical Bochner-Kähler metric on certain weighted projective spaces, and also to explicitly construct a family Kahler edge-cone metrics on $\mathbb{CP}^2$, with singular set $\mathbb{CP}^1$, having cone angles $2πβ$ for all $β>0$. We conclude by discussing how these results can be used to obtain certain well-known Einstein metrics.

math.DG↗

arXiv2015

An equivalence of scalar curvatures on Hermitian manifolds

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu-Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of noncompact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

math.DG↗