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Elliptic Complexes on Manifolds with Boundary

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Abstract

We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah–Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.

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Notes

  1. The concept of negative type can be found in [10, 11], for example.

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Authors and Affiliations

  1. Universität Potsdam, Institut für Mathematik, Potsdam, Germany

    B.-W. Schulze

  2. Dipartimento di Matematica, Università di Torino, Turin, Italy

    J. Seiler

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  1. B.-W. Schulze
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  2. J. Seiler
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Correspondence to J. Seiler.

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Schulze, BW., Seiler, J. Elliptic Complexes on Manifolds with Boundary. J Geom Anal 29, 656–706 (2019). https://doi.org/10.1007/s12220-018-0014-6

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