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Relative index theory. (English) Zbl 0762.58026

The author studies pairs of generalized Dirac operators on complete noncompact Riemannian manifolds which agree outside compact sets. By an analytic interpretation of the difference of the integrals over their local index densities, he obtains a generalization of the relative index theorem of M. Gromov and H. B. Lawson jun. [Publ. Math., Inst. Hautes Etud. Sci. 58, 295-408 (1983; Zbl 0538.53047)]. Furthermore, under some assumptions on the curvature of the manifolds, the author proves that the pair of generalized Dirac operators defines a supersymmetric scattering theory in the sense of N. V. Borisov, W. Müller and R. Schrader [Commun. Math. Phys. 114, No. 3, 475-513 (1988; Zbl 0663.58032)] and establishes a relation between the scattering index and the relative topological index.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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[1] Gromov, M; Lawson, B, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. IHES, 58, 83-196, (1983) · Zbl 0538.53047
[2] Borisov, N.V; Müller, W; Schrader, R, Relative index theorems and super-symmetric scattering theory, Comm. math. phys., 114, 475-513, (1988) · Zbl 0663.58032
[3] Donnelly, H, Essential spectrum and heat kernel, J. funct. anal., 75, 326-381, (1987) · Zbl 0634.58031
[4] Donnelly, H, Spectral geometry for certain noncompact Riemannian manifolds, Math. Z., 169, 63-76, (1979) · Zbl 0432.58022
[5] Donnelly, H; Li, P, Lower bounds for the eigenvalues of Riemannian manifolds, Michigan math. J., 29, 149-161, (1982) · Zbl 0488.58022
[6] Cheeger, J; Gromov, M; Taylor, M, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. differential geom., 17, 15-53, (1982) · Zbl 0493.53035
[7] Cheng, S.Y; Li, P; Yay, T, Upper estimate of the heat kernel, Amer. J. math., 103, 1021-1063, (1981) · Zbl 0484.53035
[8] Eichhorn, J, Elliptic differential operators on noncompact manifolds, () · Zbl 0683.58045
[9] Julg, P, Indice relatif et K-théorie bivariante de kasparov, C.R. acad. sci. Paris, 307, 243-248, (1988) · Zbl 0649.46058
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