Aaron Naber

Noncompact shrinking four solitons with nonnegative curvature

Abstract We prove that if (M, g, X) is a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M, g) is isometric to or a finite quotient of or S 3 × ℝ. In the process we also show that a complete shrinking soliton (M, g, X) with bounded curvature is gradient and κ-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc ≧ 0. The proofs rely on the technical construction of a singular reduced length function, a function which behaves as the reduced length function but can be extended to singular times.

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the

Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs

p

Topology and –regularity theorems on collapsed manifolds with Ricci curvature bounds

p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator

Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

\Delta _p

Regularity of Einstein manifolds and the codimension 4 conjecture

Δp when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.