Communications in Analysis and Geometry
Volume 28 (2020)
The First of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday
Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau
The KW equations and the Nahm pole boundary condition with knots
Pages: 871 – 942
It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a fourdimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are defined by a “Nahm pole” away from the knot with a further singularity along the knot. In a previous paper, we gave a precise formulation of the Nahm pole boundary condition in the absence of knots; in the present paper, we do this in the more general case with knots included. We show that the KW equations with generalized Nahm pole boundary conditions are elliptic, and that the solutions are polyhomogeneous near the boundary and near the knot, with exponents determined by solutions of appropriate indicial equations. This involves the analysis of a “depth two incomplete iterated edge operator.” As in our previous paper, a key ingredient in the analysis is a convenient new Weitzenböck formula that is well-adapted to the specific problem.
The research of R. Mazzeo was supported in part by NSF Grant DMS-1608223.
The research of E. Witten was supported in part by NSF Grant PHY-1606531.
Received 4 December 2017
Accepted 15 September 2018
Published 1 October 2020