Riemann surface foliations with non-discrete singular set
Sahil Gehlawat
TL;DR
This work extends the theory of leafwise Poincaré metric regularity for singular Riemann surface foliations to the non-discrete singular-set setting by introducing removable singularity hierarchies $A_l$ and strong removability $B_l$ via invariant submanifolds. It shows that under NCP and transversal-type or invariant-hypersurface conditions, the leafwise uniformization modulus $\\eta$ remains continuous on $M\\setminus E$, and, under additional discreteness or totally real assumptions, that $\\eta$ extends continuously to all of $M$. Key contributions include structural results on the $A_l$-sets (Theorem T:Removable Singular Sets) and regularity/extension theorems (Theorems T:Regularity of metric, T:Mix Regularity, T:Cont-Extn) that generalize prior discrete-$E$ findings. The findings illuminate how transversal-type geometry and nested invariant submanifolds govern transverse regularity and extension phenomena for the leafwise Poincaré metric, with implications for foliation dynamics and complex-analytic geometry.
Abstract
Let $\mathcal{F}$ be a singular Riemann surface foliation on a complex manifold $M$, such that the singular set $E \subset M$ is non-discrete. We study the behavior of the foliation near the singular set $E$, particularly focusing on singular points that admit invariant submanifolds (locally) passing through them. Our primary focus is on the singular points that are removable singularities for some proper subfoliation. We classify singular points based on the dimension of their invariant submanifold and, consequently, establish that for hyperbolic foliations $\mathcal{F}$, the presence of such singularities ensures the continuity of the leafwise Poincaré metric on $M \setminus E$.