Equivalence between A Harmonic Form and A Closed Co-Closed Form in Both Lq and Non-Lq Spaces

Authors

• Lina Wu, The City University of New York at Borough of Manhattan Community CollegeFollow

Document Type

Article

Publication Date

Spring 2-27-2017

Abstract

For a differential k-form ω on a complete non-compact manifold, we establish an equivalent relation between a harmonic form and a closed co-closed form. We extend this equivalence from ω in L^2 spaces to ω with 2-balanced growth including L^2 spaces and non-L^2 spaces. Especially for a simple differential k-form ¯ω on a complete non-compact manifold, we generalize this equivalence from ¯ω in L^q spaces to ¯ω with 2-balanced growth including L^q spaces and non-L^q spaces for 2 ≤ q < 3. Our research findings recapture the work of Andreotti and Vesentini. Our ideas and calculation methods in this paper could provide a new way of broadening L^q spaces to non-L^q spaces in a variety of energy for differential forms.

Comments

The research work was supported by PSC-CUNY Grant. Support for this project was provided by PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.