On the $L^p$ index of spin Dirac operators on conical manifolds

Details On the $L^p$ index of spin Dirac operators on conical manifolds On the $L^p$ index of spin Dirac operators on conical manifolds

2004-07-02

arxiv.org/abs/math/0407027v1

Studia Math. 177 (2006), 97-112.

Mathematics

Differential Geometry

17 pages, no figures

Scientific paper

We compute the index of the Dirac operator on spin Riemannian manifolds with conical singularities, acting from $L^p(\Sigma^+)$ to $L^q(\Sigma^-)$ with $p,q>1$. When $1+\frac{n}{p}-\frac{n}{q}>0$ we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at $\frac{n+1}{2}-\frac{n}{q}$ instead of 0 in the definition of the eta invariant. In particular we reprove Chou's formula for the $L^2$ index. For $1+\frac{n}{p}-\frac{n}{q}\leq 0$ the index formula contains an extra term related to the Calder\'on projector.

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