On \(L^ 2\)-index theorems for complete manifolds of rank-one type
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 \(g=dy^ 2\oplus g_ y\) and \(g_ y\), \(y\geq 0\), a smooth family of metrics on the compact manifold \(\partial M_ 1\).
scientific article; zbMATH DE number 78540
scientific article