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http://dx.doi.org/10.4134/JKMS.j160421

L2 HARMONIC FORMS ON GRADIENT SHRINKING RICCI SOLITONS  

Yun, Gabjin (Department of Mathematics Myong Ji University)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.4, 2017 , pp. 1189-1208 More about this Journal
Abstract
In this paper, we study vanishing properties for $L^2$ harmonic 1-forms on a gradient shrinking Ricci soliton. We prove that if (M, g, f) is a complete oriented noncompact gradient shrinking Ricci soliton with potential function f, then there are no non-trivial $L^2$ harmonic 1-forms which are orthogonal to df. Second, we show that if the scalar curvature of the metric g is greater than or equal to (n - 2)/2, then there are no non-trivial $L^2$ harmonic 1-forms on (M, g). We also show that any multiplication of the total differential df by a function cannot be an $L^2$ harmonic 1-form unless it is trivial. Finally, we derive various integral properties involving the potential function f and $L^2$ harmonic 1-forms, and handle their applications.
Keywords
gradient shrinking Ricci solitons; $L^2$ harmonic forms;
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