Abstract: For a compact manifold with boundary, M, there are well known local boundary conditions that make the de Rham operator A = d+δ elliptic, namely the absolute and relative boundary conditions. We study the eta invariants of such elliptic boundary value problems under the metric deformation
where x ɛ C
∞(M) is, near the boundary, the geodesic distance to the boundary, and g is a Riemannian metric on M which is of product type near the boundary. Under certain acyclicity condition we show that when M is odd dimensional
where the subscript a (r) indicates the absolute (relative) boundary condition, and η
b(A
0) is the b-eta invariant of the limiting operator A
0. If M is even dimensional then
Most of the analysis extends to analytic torsion, yielding
when dim M is odd, and
when dim M is even. Here the sign ± depends on the choice of the boundary condition and r
1 , r
2 vanishes at ɛ = 0.
Dedication: Dedicated to Krzysztof P. Wojciechowski on his 50th birthday
Keywords: AMSC: Primary 58J28, Secondary 35J25