Local and global scalar curvature rigidity of Einstein manifolds

On the Scalar Curvature of Einstein Manifolds

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.

On the Local Rigidity of Einstein Manifolds with Convex Boundary

Let (M, g) be a compact Einstein manifold with non-empty boundary ∂M . We prove that Killing fields at ∂M extend to Killings fields of (any) (M, g) provided ∂M is (weakly) convex and π1(M,∂M) = {e}. This gives a new proof of the classical infinitesimal rigidity of convex surfaces in Euclidean space and generalizes the result to Einstein metrics of any dimension.

On the Smooth Rigidity of Almost-einstein Manifolds with Nonnegative Isotropic Curvature

Let (Mn, g), n ≥ 4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0 < l ≤ L, we prove that there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g satisfies l ≤ s ≤ L and the Einstein tensor satisfies |Ric − s n g| ≤ ε then M is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result...

On Einstein Manifolds of Positive Sectional Curvature

Let (M,g) be a compact oriented 4-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, with its standard Fubini-Study metric.

Arithmetic Manifolds of Positive Scalar Curvature