2008/06/29 by H. Blaine Lawson, Lawson, H. Blaine, John Wermer +1
Engineering · Mathematics · Physics and Astronomy · #30H05 #32A38 #32Q99 #Advanced Differential Geometry Research #Complex Variables (math.CV) #Differential Geometry (math.DG) #Elasticity and Wave Propagation #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.0806.4776
openalex publication_date 2008/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = Cn, an answer to this question involves the polynomial hull of gamma. When X = Pn, complex projective space, the projective hull hatc of c comes into play. One always has V contained in hatc, and for analytic curves they conjecturally coincide. In this paper we establish an approximate analogue of this idea which holds without the analyticity of c. We characterize points in hatc as those which lie on a sequence of analytic disks whose boundaries converge down to c. This is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose work is essential here. The results are applied to construct a remarkable example of a closed curve c in P2, which is real analytic at all but one point, and for which the closure of hatc is W ∪ L where L is a projective line and W is an analytic (non-algebraic) subvariety of P2 - L. Furthermore, hatc itself is the union of W with only two points on L.