Holomorphic Motion of Circles through Affine Bundles

The pivotal topic of this paper is the study of Levi-flat real hypersurfaces S with circular fibers in a rank 1 affine bundle A over a Riemann surface X. (To say that S is Levi-flat is to say that S admits a foliation by Riemann surfaces; equivalently, in the language of [SuTh], S may be said to prescribe a holomorphic motion of circles through A.) After setting notation and terminology in §2 we proceed in §3 to examine the Levi-form of a general real hypersurface with circular fibers, emphasizing the connection with curvature considerations. In §4 we focus on the Levi-flat case. In Theorems 5 and 6 we construct moduli spaces for Levi-flat S attached to a fixed underlying line bundle L in the compact and non-compact cases, respectively. In particular, whenX is compact we show that the existence of a Levi-flat S implies that 0 ≤ degL ≤ 2 genus(X)− 2. (The bound is sharp.) Theorem 7 in §7 states that when S is Levi-flat, the Levi-foliation on S extends to a holomorphic foliation of the CP bundle obtained from A by compactifying the fibers. In the general case, the extended foliation in constructed by looking for holomorphic sections of A whose distance from the center is harmonic with respect to the appropriate metric. In §7 we show that this construction produces a foliation even in some cases where S “disappears into the recomplexification of A.” §6 looks at general holomorphic foliations (transverse to fibers) of compactified rank 1 affine bundles; in particular, it is shown that such foliations are classified up to equivalence by a “Schwarzian derivative” and a “curvature function.” An Addendum to Theorem 7 shows how to recognize when such a foliation arises from a Levi-flat hypersurface. The remaining sections contain postponed proofs.