Lower volume growth and total σk-scalar curvature estimates

σk-SCALAR CURVATURE AND EIGENVALUES OF THE DIRAC OPERATOR

On a 4-dimensional closed spin manifold (M, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ2-scalar curvature of M 4 as follows λ 4 ≥ 32 3 R M4 σ2(g)dvol(g) vol(M, g) . Equality implies that (M, g) is a round sphere and the corresponding eigenspinors are Killing spinors. Dedicated to Professor Wang Guangyin on the occasion of his 80th birthday

Lower Volume Growth Estimates for Self-shrinkers of Mean Curvature Flow

We obtain a Calabi-Yau type volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.

Scalar Curvature Estimates by Parallel Alternating Torsion

We generalize Llarull’s scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on ΛTM . As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups ...

Curvature Estimates in Asymptotically Flat Manifolds of Positive Scalar Curvature

Suppose that (Mn, g) is an asymptotically flat Riemannian spin manifold of positive scalar curvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifold is always positive, and is zero if and only if the manifold is flat. This result suggests that there should be an inequality which bounds the Riemann tensor in terms of the total mass and implies that curvature must ...

Curvature Estimates in Asymptotically At Manifolds of Positive Scalar Curvature