Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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L2 HARMONIC FORMS ON GRADIENT SHRINKING RICCI SOLITONS

  • Yun, Gabjin (Department of Mathematics Myong Ji University)
  • Received : 2016.06.10
  • Published : 2017.07.01
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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.

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