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On the Stability of Riemannian Manifold with Parallel Spinors Xianzhe Dai
 

Summary: On the Stability of Riemannian Manifold with Parallel Spinors
Xianzhe Dai
Xiaodong Wang
Guofang Wei
Dedicated to Jeff Cheeger for his sixtieth birthday
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.
1 Introduction
One of the most fruitful approaches to finding the `best' (or canonical) metric on a manifold has
been through the critical points of a natural geometric functional. In this approach one is led to
the study of variational problems and it is important to understand the stability issue associated

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics