On the stability of Riemannian manifold with parallel spinors
The authors study the stability problem for the total scalar curvature functional restricted to the transversal directions, i.e., the space of conformal structures. They show that Riemannian manifolds with nonzero parallel spinors (which are necessarily Ricci flat) are stable when one relates this problem to the second variation of the total scalar curvature functional restricted to the traceless symmetric \(2\)-tensors. Theorem 1.1: If a compact Riemannian manifold \((M,g)\) has a cover which is spin and admits nonzero parallel spinors, then the Lichnerowicz Laplacian \(\mathbb{L}_g\) is positive semi-definite. Theorem 1.2: Let \((M,g)\) be a compact manifold which admits a spin cover with nonzero parallel spinors. Then \(g\) cannot be deformed to positive scalar curvature metrics. Theorem 1.3: Any scalar flat deformation of a Calabi-Yau metric on a compact manifold must be Calabi-Yau. In fact, any deformation with nonnegative scalar curvature of a Calabi-Yau metric on a compact manifold is necessarily a Calabi-Yau deformation. The same is true for the other special holonomy metrics (hyper-Kähler, \(G_2\), and Spin (7)). Theorem 1.4: The Calabi-Yau (and other special holonomy) metrics are local maxima for the Yamabe invariant.
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