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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
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