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Self‐adjoint elliptic operators and manifold decompositions part I: Low Eigenmodes and stretching
Author(s) -
Cappell Sylvain E.,
Lee Ronnie,
Miller Edward Y.
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199608)49:8<825::aid-cpa3>3.0.co;2-a
Subject(s) - mathematics , symplectic geometry , pure mathematics , elliptic operator , invariant (physics) , operator (biology) , manifold (fluid mechanics) , limiting , mathematical analysis , mathematical physics , mechanical engineering , biochemistry , chemistry , repressor , transcription factor , engineering , gene
This paper is the first of a three‐part investigation into the behavior of analytical invariants of manifolds that can be split into the union of two submanifolds. In this article, we will show how the low eigensolutions of a self‐adjoint elliptic operator over such a manifold can be studied by a splicing construction. This construction yields an approximated solution of the operator whenever we have two L 2 ‐solutions on both sides and a common limiting value of two extended L 2 ‐solutions. In Part II, the present analytic “Mayer‐Vietoris” results on low eigensolutions and further analytic work will be used to obtain a decomposition theorem for spectral flows in terms of Maslov indices of Lagrangians. In Part III after comparing infinite‐ and finite‐dimensional Lagrangians and determinant line bundles and then introducing “canonical perturbations” of Lagrangian subvarieties of symplectic varieties, we will study invariants of 3‐manifolds, including Casson's invariant. © 1996 John Wiley & Sons, Inc.