Quasi-linear elliptic differential equations for mappings of manifolds, II

Gerasim Kokarev; Sergei Kuksin

doi:10.1007/s10455-006-9034-8

Quasi-linear elliptic differential equations for mappings of manifolds, II

Gerasim Kokarev, Sergei Kuksin

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel-Whitney class of such a moduli space is given by the Z₂-spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse-Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results. © Springer Science + Business Media B.V. 2007.

Original language	English
Pages (from-to)	59-113
Number of pages	55
Journal	Annals of Global Analysis and Geometry
Volume	31
Issue number	1
DOIs	https://doi.org/10.1007/s10455-006-9034-8
Publication status	Published - Jan 2007

Keywords

Elliptic equations
Infinite-dimensional moduli spaces
Morse-Bott theory
Orientation
Spectral flow

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10.1007/s10455-006-9034-8

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abstract = "We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel-Whitney class of such a moduli space is given by the Z2-spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse-Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results. {\textcopyright} Springer Science + Business Media B.V. 2007.",

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AU - Kuksin, Sergei

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AB - We study questions related to the orientability of the infinite-dimensional moduli spaces formed by solutions of elliptic equations for mappings of manifolds. The principal result states that the first Stiefel-Whitney class of such a moduli space is given by the Z2-spectral flow of the families of linearised operators. Under an additional compactness hypotheses, we develop elements of Morse-Bott theory and express the algebraic number of solutions of a non-homogeneous equation with a generic right-hand side in terms of the Euler characteristic of the space of solutions corresponding to the homogeneous equation. The applications of this include estimates for the number of homotopic maps with prescribed tension field and for the number of the perturbed pseudoholomorphic tori, sharpening some known results. © Springer Science + Business Media B.V. 2007.

KW - Elliptic equations

KW - Infinite-dimensional moduli spaces

KW - Morse-Bott theory

KW - Orientation

KW - Spectral flow

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DO - 10.1007/s10455-006-9034-8

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EP - 113

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

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Quasi-linear elliptic differential equations for mappings of manifolds, II

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