Self-adjoint elliptic operators and manifold decompositions Part I: Low eigenmodes and stretching

Sylvain E. Cappell, Ronnie Lee, Edward Y. Miller

Research output: Contribution to journalArticle

Abstract

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 L2-solutions on both sides and a common limiting value of two extended L2 -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.

Original languageEnglish (US)
Pages (from-to)825-866
Number of pages42
JournalCommunications on Pure and Applied Mathematics
Volume49
Issue number8
DOIs
StatePublished - Aug 1996

Fingerprint

Self-adjoint Operator
Elliptic Operator
Stretching
Decomposition
Decompose
Casson Invariant
Maslov Index
Spectral Flow
Invariant
Decomposition Theorem
Line Bundle
Submanifolds
Determinant
Union
Limiting
Perturbation
Operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Self-adjoint elliptic operators and manifold decompositions Part I : Low eigenmodes and stretching. / Cappell, Sylvain E.; Lee, Ronnie; Miller, Edward Y.

In: Communications on Pure and Applied Mathematics, Vol. 49, No. 8, 08.1996, p. 825-866.

Research output: Contribution to journalArticle

@article{6699c904aa8842cbb75ea87689b8bd53,
title = "Self-adjoint elliptic operators and manifold decompositions Part I: Low eigenmodes and stretching",
abstract = "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 L2-solutions on both sides and a common limiting value of two extended L2 -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.",
author = "Cappell, {Sylvain E.} and Ronnie Lee and Miller, {Edward Y.}",
year = "1996",
month = "8",
doi = "10.1002/(SICI)1097-0312(199608)49:8<825::AID-CPA3>3.3.CO;2-4",
language = "English (US)",
volume = "49",
pages = "825--866",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "8",

}

TY - JOUR

T1 - Self-adjoint elliptic operators and manifold decompositions Part I

T2 - Low eigenmodes and stretching

AU - Cappell, Sylvain E.

AU - Lee, Ronnie

AU - Miller, Edward Y.

PY - 1996/8

Y1 - 1996/8

N2 - 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 L2-solutions on both sides and a common limiting value of two extended L2 -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.

AB - 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 L2-solutions on both sides and a common limiting value of two extended L2 -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.

UR - http://www.scopus.com/inward/record.url?scp=0030494290&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030494290&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1097-0312(199608)49:8<825::AID-CPA3>3.3.CO;2-4

DO - 10.1002/(SICI)1097-0312(199608)49:8<825::AID-CPA3>3.3.CO;2-4

M3 - Article

AN - SCOPUS:0030494290

VL - 49

SP - 825

EP - 866

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 8

ER -