On \(L^ 2\)-index theorems for complete manifolds of rank-one type
type: Item
instance of: scholarly article
title: On \(L^ 2\)-index theorems for complete manifolds of rank-one type
zbMATH Open document ID: 0765.58030
DOI: 10.1215/S0012-7094-92-06608-7
published in: Duke Mathematical Journal
publication date: 1992-12-14T00:00:00Z
review text: The author continues his investigations on the index problem for elliptic differential operators on noncompact manifolds [J. Differ. Geom. 32, No. 2, 491-532 (1990; Zbl 0722.58043)]. He generalizes his methods used in the former paper to get results in a more general setting. Let \(M\) be a Riemannian manifold and \(E_ 1\), \(E_ 2\) two Hermitean vector bundles over \(M\), and let \(D:C^ \infty(E_ 1)\to C^ \infty(E_ 2)\) be a first order elliptic differential operator. If \(M\) is noncompact it may occur that no extension of \(D\) is a Fredholm operator but that \(D\) has a finite \(L^ 2\)-index in the sense that \(\text{ker} D\cap L^ 2(E_ 1)\) and \(\text{ker} D'\cap L^ 2(E_ 2)\) both have finite dimension, \(D'\) denotes the formal adjoint of \(D\). Then the \(L^ 2\)-index of \(D\) is defined as \[ L^ 2\text{-ind} D=\dim\text{ker} D\cap L^ 2(E_ 1)- \dim\text{ker} D'\cap L^ 2(E_ 2). \] It is assumed that there exists an open subset \(U\) of \(M\) with the following properties: \(M_ 1:=M\backslash U\) is a compact manifold with boundary and the operator \(D\) has a representation over \(U\) as an operator in \(L^ 2((0,\infty),H)\), where \(H=L^ 2(F)\) for some Hermitean vector bundle \(F\) over \(\partial M_ 1\). For any \(u\in C_ 0^ \infty([0,\infty),H)\) the operation of \(D\) on \(u\) is given by \(-\partial_ yu(y)+S(y)u(y)\), where \(S(y)\), \(y\geq 0\), is a smooth family of self adjoint first order elliptic differential operators acting on \(C^ \infty(F)\). Under some additional assumptions the author proves that \(D\) has finite \(L^ 2\)- index, and he is able to deduce an index formula for the \(L^ 2\)-index. The formula contains interior terms, coming from the operator over \(M_ 1\), spectral invariants coming from the operator \(S\) and some global invariants. As an application the signature operator on a complete Riemannian manifold is studied, where \(M=M_ 1\cup U\) and \(U\) is product like, i.e. \(U=(0,\infty)\times\partial M_ 1\) with metric \(g=dy^ 2…
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 78540
zbMATH Keywords: index theorems on noncompact manifolds
zbMATH Keywords: \(L^ 2\)-index
author: Jochen Brüning
reviewed by: K. H. Mayer
MaRDI profile type: Publication
cites work: Eta invariants, signature defects of cusps, and values of L-functions
cites work: Spectral asymmetry and Riemannian Geometry. I
cites work: L\({}^ 2\)-index theorems on certain complete manifolds
cites work: The resolvent expansion for second order regular singular operators
cites work: An Index Theorem for First Order Regular Singular Operators
cites work: \(L^ 2\)-index and the Selberg trace formula
cites work: Q5526483
cites work: Axial anomalies and index theorems on open spaces
cites work: Eta invariant of a fibered manifold
cites work: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds
cites work: Q3679959
cites work: Q5526485
cites work: Defect subspaces and generalized resolvents of an Hermitian operator in the space \(\Pi_\kappa\)
cites work: Q3995493
cites work: Signature defects of cusps of Hilbert modular varieties and values of L- series at \(s=1\)
cites work: Manifolds with cusps of rank one. Spectral theory and \(L^ 2\)-index theorem
cites work: Q3805508
cites work: 𝐿²-index of elliptic operators on locally symmetric spaces of finite volume
cites work: Extending \({\bar \partial}\) to singular Riemann surfaces
cites work: \(L^ 2\)-index theorems on locally symmetric spaces
cites work: Spectral theory of ordinary differential operators
description: scientific article; zbMATH DE number 78540