×

Fibered cusp versus \(d\)-index theory. (English) Zbl 1142.53039

Let \(\overline{X}\) be a compact manifold-with-boundary and let \(\pi : \partial X \to Y\) be a locally trivial bundle whose base is a closed manifold. Let \(\rho\) be a defining function for \(\partial X\). Let \(T(X)^\Phi\) be the \(\Phi\)-tangent bundle over \(\overline{X}\), so that the restriction to \(\partial X\) of any smooth section \(V : \overline{X} \to T(X)^\Phi\) is vertical and \(V\) vanishes at \(\partial X\) to the second order. Let \(g_\Phi\) be a fibered cusp metric i.e. \(g_\Phi\) is the restriction to \(X = \overline{X} \setminus \partial X\) of a Riemannian metric in \(T(X)^\Phi\) which is smooth up to the boundary. One calls \(g_\Phi\) exact if there is a tensor field \(S \in C^\infty (\overline{X} , S^2 (T(X)^\Phi ))\) such that \(g_\Phi - \rho \, S\) is a product \(\Phi\)-metric, i.e., a \(\Phi\)-metric of the form \(\rho^{-4} \, d \rho^2 + \rho^{-2} \pi^* g_Y + h\) near infinity, where \(g_Y\) is a Riemannian metric on \(Y\) and \(h\) is a family of metrics on the fibres of \(\pi\). To state the main result in the paper under review let \(g_\Phi\) be an exact \(\Phi\)-metric on \(X\). Let \(E \to \overline{X}\) be a Hermitian vector bundle with connection smooth up to the boundary. Let \(X\), \(\partial X\) and the fibres of \(\pi\) be endowed with compatible spin structures. Let \(D^\Phi\) and \(D^p\) be the Dirac operators associated to \(g_\Phi\) and \(\rho^{2p} g_\Phi\) for \(p \geq 0\). Let us assume that that the twisted Dirac operators on the fibres of the portion of \(E\) over \(\partial X\), the induced metric on the fibres of \(\pi\), and the induced spin structure are invertible. Then multiplication by \(\rho^{p(n-1)/2}\) gives an isomorphism of the \(L^2\)-kernels of \(D^p\) and \(D^\Phi\).
Reviewer’s remark: For any compact strictly pseudoconvex CR manifold-with-boundary \(M\) the total space \(\overline{X}\) of the canonical circle bundle \(S^1 \to \overline{X} \to M\) over \(M\) is a compact manifold-with-boundary such that \(S^1 \to \partial X \to \partial M\) is a principal bundle. It is an appealing question whether these results apply to \(\overline{X}\) endowed with the Fefferman metric in the context of the spinor calculus developed by H. Baum [Dragomir, Sorin (ed.), Selected topics in Cauchy-Riemann geometry. Rome: Aracne. Quad. Mat. 9, 39–87 (2003; Zbl 1066.53095)]

MSC:

53C27 Spin and Spin\({}^c\) geometry
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C80 Applications of global differential geometry to the sciences
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)

Citations:

Zbl 1066.53095
PDFBibTeX XMLCite
Full Text: arXiv EuDML

References:

[1] J.-M. BISMUT - J. CHEEGER, h-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), pp. 33-70. Zbl0671.58037 MR966608 · Zbl 0671.58037 · doi:10.2307/1990912
[2] E. LEICHTNAM - R. MAZZEO - P. PIAZZA, The index of Dirac operators on manifolds with fibered boundaries, to appear in Proc. Joint BeNeLuxFra Confer. Math., Ghent, May 20-22, 2005, Bull. Belg. Math. Soc. - Simon Stevin. Zbl1126.58009 MR2293212 · Zbl 1126.58009
[3] R. R. MAZZEO - R. B. MELROSE, Pseudodifferential operators on manifolds with fibered boundaries, Asian J. Math., 2 (1998), pp. 833-866. Zbl1125.58304 MR1734130 · Zbl 1125.58304
[4] R. B. MELROSE, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics 4, A. K. Peters, Wellesley, MA (1993). Zbl0796.58050 MR1348401 · Zbl 0796.58050
[5] R. B. MELROSE - F. ROCHON, Index in K-theory for families of fibred cusp operators, preprint math. DG/0507590. Zbl1126.58010 · Zbl 1126.58010 · doi:10.1007/s10977-006-0003-6
[6] S. MOROIANU, Weyl laws on open manifolds, preprint math. DG/0310075. Zbl1131.58019 MR2349766 · Zbl 1131.58019 · doi:10.1007/s00208-007-0137-8
[7] W. MÜLLER, Manifolds with cusps of rank one. Spectral theory and L2 -index theorem, Lecture Notes in Math. 1244, Springer-Verlag, Berlin, 1987. Zbl0632.58001 MR891654 · Zbl 0632.58001
[8] B. VAILLANT, Index- and spectral theory for manifolds with generalized fibered cusps, Dissertation, Bonner Math. Schriften 344 (2001), Rheinische Friedrich-Wilhelms-Universität Bonn. Zbl1059.58018 MR1933455 · Zbl 1059.58018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.