Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I

TL;DR

This paper provides a detailed, self-contained construction of Kuranishi structures on moduli spaces of pseudo holomorphic disks, utilizing exponential decay estimates, and corrects previous typographical errors in the foundational lemmas and propositions.

Contribution

It offers an improved, comprehensive construction of Kuranishi structures on moduli spaces of pseudo holomorphic disks, building on and correcting earlier work.

Findings

Completed the construction of Kuranishi structures for a single moduli space.

Corrected typographical errors in key lemmas and propositions.

Enhanced the clarity and self-containment of the construction process.

Abstract

This is the first of two articles in which we provide detailed and self-contained account of the construction of a system of Kuranishi structures on the moduli spaces of pseudo holomorphic disks, using the exponential decay estimate given in [FOOO7]. This article completes the construction of a Kuranishi structure of a single moduli space. This article is an improved version of [FOOO4, Part 4] and its mathematical content is taken from our earlier writing [FOn,FOOO2,FOOO4,FOOO7]. The version 2 contains a correction of typological errors in the statement of Lemma 10.11 and the statement of Proposition 10.4. The proof of them are not changed. (Note these places were written in the same way as version 1 in the version published in Surveys in Differential Geometry.)