Abstract
In this paper we extend the results obtained in [9], [10] to manifolds with Spinℂ-structures defined, near the boundary, by an almost complex structure. We show that on such a manifold with a strictly pseudoconvex boundary, there are modified ∂¯-Neumann boundary conditions defined by projection operators, eo+, which give subelliptic Fredholm problems for the Spinℂ-Dirac operator, ðeo+. We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of (ðeo+,eo+) equals the relative index, on the boundary, between eo+ and the Calderón projector, eo+. Using the relative index formalism, and in particular, the comparison operator, eo+, introduced in [9], [10], we prove a trace formula for the relative index that generalizes the classical formula for the index of an elliptic operator. Let (X0,J0) and (X1,J1) be strictly pseudoconvex, almost complex manifolds, with ϕ:bX1→bX0, a contact diffeomorphism. Let 0,1 denote generalized Szegő projectors on bX0,bX1, respectively, and eo0, eo1, the subelliptic boundary conditions they define. If X1⎯⎯⎯⎯ is the manifold X1 with its orientation reversed, then the glued manifold X=X0⨿ϕX1⎯⎯⎯⎯ has a canonical Spinℂ-structure and Dirac operator, ðeoX. Applying these results and those of our previous papers we obtain a formula for the relative index, R−Ind(0,ϕ∗1),
R−ind(0,ϕ∗1)=Ind(ðeX)−Ind(ðeX0,e0)+Ind(ðeX1,e1).
For the special case that X0 and X1 are strictly pseudoconvex complex manifolds and 0 and 1 are the classical Szegő projectors defined by the complex structures this formula implies that
R−ind(0,ϕ∗1)=Ind(ðeX)−χ′(X0)+χ′(X1),
which is essentially the formula conjectured by Atiyah and Weinstein; see [37]. We show that, for the case of embeddable CR-structures on a compact, contact 3-manifold, this formula specializes to show that the boundedness conjecture for relative indices from [7] reduces to a conjecture of Stipsicz concerning the Euler numbers and signatures of Stein surfaces with a given contact boundary; see [35].
ABOUT THE AUTHOR
数学年刊(Annals of Mathematics)
0 Following
0 Fans
0 Projects
674 Articles
SIMILAR ARTICLES
Abstract
For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin
Read More
Abstract
For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we
Read More
Abstract
Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let
Read More
Abstract
We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely
Read More
Abstract
We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We
Read More
Abstract
This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal
Read More
Abstract
We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge
Read More
Abstract
We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so
Read More
Abstract
We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat
Read More
Abstract
If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt
Read More