Atiyah’s $L^2$-index theorem relates the index of an elliptic operator on a
closed manifold to the $L^2$-index of the elliptic operator lifted to a covering
space. The theorem is useful in the study of existence of nontrivial solutions
for elliptic PDE on noncompact manifolds, for example, $L^2$-harmonic forms. In
this talk, we consider a type of noncompact manifolds, so-called manifolds with
regular exhaustion, introduced by Roe, and show that Atiyah’s $L^2$-index
theorem still hold. This is joint work with Guoliang Yu and Dapeng Zhou.