## Addendum to: “Subelliptic Spin$$_{\mathbb{C}}$$ Dirac operators. III”.(English)Zbl 1256.32034

In this addendum to [Ann. Math. (2) 168, No. 1, 299–365 (2008; Zbl 1169.32008)] the author proves the relative index conjecture. Using this result the author proves that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the $$C^\infty$$-topology.
In particular, let $$Y$$ be a 3 dimensional manifold and $$H \subset TY$$ a plane field defining a contact structure. We can define a strictly pseudoconvex CR-structure on $$Y$$ by a complex structure on $$H$$. The manifold is fillable if it can be embedded into some $${\mathbb{C}}^N$$ as a boundary of a normal Stein space. A Szegő projector is the $$L^2$$-orthogonal projection onto the $$L^2$$-closure of $$\ker \bar{\partial}_b$$. If $$S_1$$ is another Szegő projector defined by a fillable deformation of the reference structure then the restriction $$S_1 : \text{Im} S_0 \to \text{Im} S_1$$ is a Fredholm operator and the Fredholm index is the relative index $$\text{R-Ind}(S_0,S_1)$$. See also [the author, Ann. Math. (2) 147, No. 1, 1–59 (1998; Zbl 0942.32025)].
In this addendum the author proves that if $$(Y,H)$$ is a compact 3-dimensional co-oriented contact manifold and $$S_0$$ the Szegő projector defined by a fillable CR-structure on $$Y$$ with underlying plane field $$H$$, then there is an $$M \geq 0$$ such that if $$S_1$$ is the Szegő projector defined by any fillable deformation of the reference structure, with the plane field $$H$$, then $$\text{R-Ind}(S_0,S_1) \leq M$$. As a corollary, the set of fillable deformations of the CR structure on $$Y$$ is closed in the $$C^\infty$$-topology.

### MSC:

 32V30 Embeddings of CR manifolds 32W10 $$\overline\partial_b$$ and $$\overline\partial_b$$-Neumann operators 32V15 CR manifolds as boundaries of domains

### Citations:

Zbl 1169.32008; Zbl 0942.32025
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### References:

 [1] F. A. Bogomolov and B. de Oliveira, ”Stein small deformations of strictly pseudoconvex surfaces,” in Birational Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1997, vol. 207, pp. 25-41. · Zbl 0889.32021 [2] C. L. Epstein, ”A relative index on the space of embeddable CR-structures. I,” Ann. of Math., vol. 147, iss. 1, pp. 1-59, 1998. · Zbl 0942.32025 [3] C. L. Epstein, ”Erratum: A relative index on the space of embeddable CR-structures. I,” Ann. of Math., vol. 154, iss. 1, pp. 223-226, 2001. · Zbl 0983.32036 [4] C. L. Epstein, ”A relative index on the space of embeddable CR-structures. II,” Ann. of Math. (2), vol. 147, iss. 1, pp. 61-91, 1998. · Zbl 0942.32026 [5] C. L. Epstein, ”Subelliptic $${ Spin}_{\mathbb C}$$ Dirac operators. III. The Atiyah-Weinstein conjecture,” Ann. of Math., vol. 168, iss. 1, pp. 299-365, 2008. · Zbl 1169.32008 [6] L. Lempert, ”On three-dimensional Cauchy-Riemann manifolds,” J. Amer. Math. Soc., vol. 5, iss. 4, pp. 923-969, 1992. · Zbl 0781.32014 [7] L. Lempert, ”Embeddings of three-dimensional Cauchy-Riemann manifolds,” Math. Ann., vol. 300, iss. 1, pp. 1-15, 1994. · Zbl 0817.32009 [8] L. Lempert, ”Algebraic approximations in analytic geometry,” Invent. Math., vol. 121, iss. 2, pp. 335-353, 1995. · Zbl 0837.32008 [9] A. I. Stipsicz, ”On the geography of Stein fillings of certain 3-manifolds,” Michigan Math. J., vol. 51, iss. 2, pp. 327-337, 2003. · Zbl 1043.53066
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