Exponential decay estimates and smoothness of the moduli space of pseudoholomorphic curves. (English) Zbl 07913536
Memoirs of the American Mathematical Society 1500. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-7106-4/pbk; 978-1-4704-7878-0/ebook). v, 138 p. (2024).
Summary: The compacified moduli space of bordered stable maps carries a Kuranishi structure with boundary. Smoothness of Kuranishi structure along the boundary requires smoothness of coordinate changes along the boundary. The proof of smoothness is written by the authors in [AMS/IP Stud. Adv. Math., 46.1, 2009, xii+396 pp.], [AMS/IP Stud. Adv. math., 46.2, 2009, pp. i-xii and 397-805], and [Surv. Differ. Geom., vol. 22, pp. 133-190, 2018] based on some uniform exponential decay estimates of the stable maps with respect to a parameter \(T\), the length of the gluing cylindrical neck, near the boundary of the moduli space. In this paper we establish this exponential decay estimates in a precise manner by carefully examining the dependence on the parameter \(T\) of the gluing construction in the setting of bordered Riemann surfaces with boundary punctures. We also show that the smoothness of the collar follows from the aforementioned exponential estimates by taking \(s = 1/T\) as the radial coordinate of the collar.
MSC:
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53D35 | Global theory of symplectic and contact manifolds |
| 58D27 | Moduli problems for differential geometric structures |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
Keywords:
pseudo-Holomorphic curve; Kuranishi structure; gluing; exponential decay; weighted Sobolev spaceReferences:
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