Dan Knopf

Estimating the trace-free Ricci tensor in Ricci flow

An important and natural question in the analysis of Ricci flow behavior in all dimensions n ≥ 4 is this: What are the weakest conditions that guarantee that a solution remains smooth? In other words, what are the weakest conditions that provide control of the norm of the full Riemann curvature tensor? In this short paper, we show that the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, for all compact solutions in all dimensions n ≥ 3, without any hypotheses on the initial data.

Local monotonicity and mean value formulas for evolving Riemannian manifolds

Abstract We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local monotone quantity directly analogous to Perelmans reduced volume V and a local identity related to Perelmans average energy F.

Formal matched asymptotics for degenerate Ricci flow neckpinches

Gu and Zhu (2008 Commun. Anal. Geom. 16 467–94) have shown that type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on . In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.

Asymptotic stability of the cross curvature flow at a hyperbolic metric

We show that for any hyperbolic metric on a closed 3-manifold, there exists a neighborhood such that every solution of a normalized cross curvature flow with initial data in this neighborhood exists for all time and converges to a constant-curvature metric. We demonstrate that the same technique proves an analogous result for Ricci flow. Additionally, we prove short-time existence and uniqueness of cross curvature flow under slightly weaker regularity hypotheses than were previously known.

POSITIVITY OF RICCI CURVATURE UNDER THE KÄHLER–RICCI FLOW

In each complex dimension n ≥ 2, we construct complete Kahler manifolds of bounded curvature and non-negative Ricci curvature whose Kahler–Ricci evolutions immediately acquire Ricci curvature of mixed sign.

Minimally invasive surgery for Ricci flow singularities

Abstract In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on 𝒮n + 1, without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelmans hope for a “canonically defined Ricci flow through singularities”.

Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow

We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are

Universality in mean curvature flow neckpinches

C^3

Ricci flow neckpinches without rotational symmetry

-close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singularity formation. Our results show in a precise way that MCF solutions become asymptotically rotationally symmetric near a neckpinch singularity.

Cross curvature flow on a negatively curved solid torus

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is C^3-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.