Degree formulas for maps with nonintegrable Jacobian
type: Item
instance of: scholarly article
title: Degree formulas for maps with nonintegrable Jacobian
zbMATH Open document ID: 0854.58005
DOI: 10.12775/TMNA.1995.033
author: Tadeusz Iwaniec
author: Carlo Sbordone
author: Luigi Greco
author: Bianca Stroffolini
published in: Topological Methods in Nonlinear Analysis
publication date: 1997-01-22T00:00:00Z
review text: The well-known formula \(\deg (f;X,Y) = \int_X f^\sharp \omega\) \((\int_Y \omega = 1)\) for the degree of a \(C^1\)-map \(f : X \to Y\) between closed \(n\)-dimensional manifolds is generalized under the additional assumption of nonvanishing homologies \(H^\ell (Y) \neq 0\) for certain \(\ell\) \((1 \leq \ell \leq n-1)\) as follows. Let \(X\), \(Y\) be equipped with smooth Riemannian structures, and \(\mu \in H^\ell (Y)\) be a harmonic field with \(\int_Y |\mu|^2 = 1\). Then the formula \[ \deg (f;X,Y) = \lim \int_X (f^\sharp \mu \mid f^\flat\mu)/|f^\sharp \mu|^\varepsilon |f^\flat\mu|^\varepsilon \qquad (\varepsilon \downarrow 0) \] generalizes the degree for each map \(f \in {\mathcal C}^n (X,Y)\). Here \({\mathcal C}^n(X,Y)\) is a certain class of mappings (which cannot be specified here) including some Sobolev-Orlicz spaces broader than \(W^{1,p} (X,Y)\), and \(f^\sharp\) is the pull-back, \(f^\flat = (-1)^{\ell n - \ell} *f^\sharp *\) the composition with the Hodge operator.
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 49Q15
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 55M25
zbMATH DE Number: 906857
zbMATH Keywords: mapping degree
zbMATH Keywords: Hodge theory
zbMATH Keywords: Jacobian
reviewed by: Jan Chrastina
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 906857