Robert Haslhofer

A Compactness Theorem for Complete Ricci Shrinkers

We prove precompactness in an orbifold Cheeger–Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss–Bonnet with cutoff argument.

Mean curvature flow with surgery

We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in

Uniqueness of the bowl soliton

R^N

Dynamical stability and instability of Ricci-flat metrics

, as announced in arXiv:1304.0926. Our proof works for all

A NOTE ON THE COMPACTNESS THEOREM FOR 4d RICCI SHRINKERS

N \geq 3

A renormalized Perelman-functional and a lower bound for the ADM-mass

, including mean convex surfaces in

Mean Curvature Flow of Mean Convex Hypersurfaces

R^3

On Brendle’s Estimate for the Inscribed Radius Under Mean Curvature Flow

. We also derive a priori estimates for a more general class of flows in a local and flexible setting.

Quantitative Stratification and the Regularity of Mean Curvature Flow

We prove that any translating soliton for the mean curvature flow which is noncollapsed and uniformly 2-convex must be the rotationally symmetric bowl soliton. In particular, this proves a conjecture of White and Wang, in the 2-convex case in arbitrary dimension.

Perelman’s lambda-functional and the stability of Ricci-flat metrics

In this short article, we improve the dynamical stability and instability results for Ricci-flat metrics under Ricci flow proved by Sesum (Duke Math J 133:1–26, 2006) and Haslhofer (Calc Var Partial Differ Equ 45:481–504, 2012), getting rid of the integrability assumption.