Subelliptic boundary conditions for Spinℂ–Dirac operators, gluing, relative indices, and tame Fredholm pairs
Subelliptic boundary conditions for Spinℂ–Dirac operators, gluing, relative indices, and tame Fredholm pairs
复制标题
DOI：
10.1073/pnas.0605368103
复制
发表时间：
2006-10
期刊：
影响因子：
--
通讯作者：
C. Epstein
中科院分区：
文献类型：
--
作者：
C. Epstein
设X是具有边界的自旋ℂ流形，使得自旋ℂ结构在边界附近由一个几乎复杂的结构定义，该结构要么是严格伪凸的，要么是伪凹的(因此是接触的)。利用推广的Szegő投影器，我们定义了旋量的修正∂̄-Neumann边界条件ℛEO，从而得到了自旋ℂ-Dirac算子的次椭圆型Fredholm型边值问题。为了研究这些边值问题的指标，我们将Fredholm偶的推广引入到“Tame”范畴。在这一背景下，我们证明了图的闭包的指数等于ℛEO和ℛ射影之间的边界上的相对指数。设X0和X1是严格伪凸的自旋ℂ流形，如上所述。设φ：bx1→bx0是一个接触微分同胚，s0，s1分别表示bx0，bx1上的广义ő投影子，ℛ0eo，ℛ1eo是它们定义的次椭圆边界条件.如果X1是方向相反的流形X1，则粘合流形X=X0∐φX1具有正则自旋ℂ结构和狄拉克算符。应用这些结果，我们得到了相对指数R-IND(S0，φ*S1)的一个公式，作为特例，这个公式验证了Atiyah和Weinstein[(1997)rims Kokyuroku1014：1-14]关于电离层丛之间接触变换的量子化指数的猜想。
Let X be a Spinℂ manifold with boundary, such that the Spinℂ structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szegő projectors, we define modified ∂̄-Neumann boundary conditions, ℛeo, for spinors, which lead to subelliptic Fredholm boundary value problems for the Spinℂ–Dirac operator, ðeo. To study the index of these boundary value problems 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, ℛeo) equals the tame relative index, on the boundary, between ℛeo and the Calderon projector. Let X0 and X1 be strictly pseudoconvex, Spinℂ manifolds, as above. Let φ : bX1 → bX0, be a contact diffeomorphism, S0, S1 denote generalized Szegő projectors on bX0, bX1, respectively, and ℛ0eo, ℛ1eo, the subelliptic boundary conditions they define. If X1 is the manifold X1 with its orientation reversed, then the glued manifold X = X0 ∐φ X1 has a canonical Spinℂ structure and Dirac operator, ðXeo. Applying these results we obtain a formula for the relative index, R-Ind(S0, φ*S1), As a special case, this formula verifies a conjecture of Atiyah and Weinstein [(1997) RIMS Kokyuroku 1014:1–14] for the index of the quantization of a contact transformation between cosphere bundles.