An Index Formula for Cauchy-Riemann Operators on Surfaces with Boundary Punctures

TL;DR

This paper provides a new, self-contained proof of the index formula for Cauchy-Riemann operators on surfaces with boundary punctures, introducing a novel weighted boundary zero count to aid future research.

Contribution

The paper introduces a self-contained proof of the index formula for Cauchy-Riemann operators on punctured surfaces, incorporating a new weighted boundary zero count.

Findings

Derived a formula for the Fredholm index involving a weighted boundary zero count.

Extended the method of large antilinear deformations to surfaces with boundary punctures.

Provided a proof that may facilitate further research in symplectic geometry and related fields.

Abstract

We give a self-contained proof of a formula computing the Fredholm index for asymptotically non-degenerate Cauchy-Riemann operators on surfaces with boundary punctures using the method of large antilinear deformations. This method for computing the index was introduced in the case of closed surfaces by Taubes and generalized to the case with interior punctures by Gerig. One novel feature arising from our proof is that the Euler characteristic term in the index formula involves a non-standard weighted count of boundary zeros. We hope that this formulation of the index formula will be useful to other researchers.