Hermitian Metrics and Singular Riemann Surface Foliations

Abstract

The main aim of this thesis is to understand curvature properties of a given hermitian metric by restricting it to the leaves of a suitable singular Riemann surface foliation. We will specifically consider the complete Kahler metric example given by Grauert on (C^n)*, from which we can construct complete Kahler metric on the complement of any closed analytic subset of a Stein manifold. Note that for n ≥ 2, (C^n)* = C^n - {0} is a non-domain of holomorphy. So, this construction helps us to get a complete Kahler metric on the complement of singular set of a Riemann surface foliation.

We will study this metric in some prototype models, i.e., we will calculate holomorphic sectional curvature of the Grauert metric (for n ≥ 2) on (i) (C^n)* and (ii) B^n - A, where A ⊂ B^n is a k-dimensional affine plane, where 0 ≤ k ≤ n-2, and make observations about how this curvature behaves near singular points and away from singular points. We will observe that this is a non-positively curved metric on C*.

Next, we will use this construction to get a better understanding of this complete Kahler metric on the complement of a non-negative principal divisor D in a Stein manifold M. We will prove some facts about its holomorphic sectional curvature in this case. We will also prove that there is an intrinsic continuity property in its construction. More specifically, we will show that we can choose these metrics in a continuous fashion if the corresponding divisor vary continuously in an appropriate sense. The same regularity can be seen for the Gaussian curvature of the restricted metric on leaves of a foliation induced by a non-singular holomorphic vector field, as the corresponding non-negative principal divisor vary continuously.

Finally, we will consider the leafwise Poincare metric of a singular Riemann surface lamination whose leaves are all hyperbolic and prove that this metric converges pointwise if the domain of definition of this lamination varies in an appropriate sense. This convergence is uniform on compacts under additional hypothesis on the domain of definition and regularity of leafwise Poincare metric of the lamination of limiting domain. We will also give an analogue of the Minda's Domain Bloch constant in the foliated case and show that a similar lower bound holds in this case also as in the case of hyperbolic Riemann surfaces.