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

Volume 177 / 2006

André Legrand, Sergiu Moroianu Studia Mathematica 177 (2006), 97-112 MSC: Primary 58J20. DOI: 10.4064/sm177-2-1

Abstract

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

Authors

  • André LegrandUFR MIG
    Université Paul Sabatier
    118 route de Narbonne
    31062 Toulouse, France
    e-mail
  • Sergiu MoroianuInstitutul de Matematică
    al Academiei Române
    P.O. Box 1-764
    RO-014700 Bucureşti, Romania
    e-mail

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