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On the stability of Riemannian manifold with parallel spinors

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Abstract

Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results.

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Authors and Affiliations

  1. Department of Mathematics, UCSB, Santa Barbara, CA, 93106, USA

    Xianzhe Dai & Guofang Wei

  2. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Xiaodong Wang

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  1. Xianzhe Dai
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  2. Xiaodong Wang
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  3. Guofang Wei
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Correspondence to Xianzhe Dai, Xiaodong Wang or Guofang Wei.

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Dedicated to Jeff Cheeger for his sixtieth birthday

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Dai, X., Wang, X. & Wei, G. On the stability of Riemannian manifold with parallel spinors. Invent. math. 161, 151–176 (2005). https://doi.org/10.1007/s00222-004-0424-x

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