Search

We survey some ideas regarding the application of the Aleksandrov reflection method in partial differential equation to extrinsic geometric flows of Euclidean hypersurfaces. In this survey, we mention some related and important recent developments of others on the convergence of noncontracting flows and construction and classification of ancient flows.

In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity theory analogous to Cheeger--Colding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of $4$-dimensional non-Ricci-flat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature., 14 pages

As indicated by the third author in [19], there is a gap in the previous version of this paper by the first two authors [5]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelman's techniques, is implied by Ding [9] and is covered by [19]. Our approach, however, is different from theirs. In addition, we prove a necessary and sufficient condition for a three-dimensional $\kappa$-solution to form a forward singularity. We hope that this condition is helpful in the classification of all three-dimensional $\kappa$-solutions. Up to now, the only main progress on such a classification, as conjectured by Perelman, is by Brendle [2].

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to $3$-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form $\mathbb{S}^3/\Gamma$ or weakly dimension reduce to the $3$-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE $4$-manifolds or dimension reduce to $3$-dimensional manifolds. As an application, we prove that any steady gradient K\"{a}hler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperk\"{a}hler ALE Ricc-flat $4$-manifolds., Comment: A volume comparison result is corrected. See the discussion in the appendix. The paper is accepted by Adv in Math

In this note we discuss estimates for the curvature of 4-dimensional gradient Ricci soliton singularity models by applying Perelman's point selection, a fundamental result of Cheeger and Naber, and topological lemmas., Comment: Revised version, as appeared in Advances in Mathematics 372 (2020), article number 107303. We would like to thank the referee for their suggestions

We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.

In this expository note we discuss how to apply the inverse function theorem to prove the folklore result that on the unit n-sphere for n ≥ 3 any harmonic self-map near the identity must be an isometry.

Following Li and Yau (Acta Math 156:153–201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional $${\mathcal{J}}$$ associated to a smooth function $${\phi}$$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.

In this purely expository note, we deduce a few known direct consequences of Perelman’s canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional κ-solutions. These corollaries regard elementary properties of 3-dimensional singularity models and κ-solutions.

(2007). Semidiscrete Geometric Flows of Polygons. The American Mathematical Monthly: Vol. 114, No. 4, pp. 316-328.

For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.

An important component of Hamilton’s program for the Ricci flow on compact 3-manifolds is the classification of singularities which form under the flow for certain initial metrics. In particular, Type I singularities, where the evolving metrics have curvatures whose maximums are inversely proportional to the time to blow-up, are modelled on the 3-sphere and the cylinder S × R and their quotients. On the other hand, Type II singularities (the complementary case) are much more difficult to understand. Despite this, it is known from the work of Hamilton that their singularity models are stationary solutions to the Ricci flow. This uses several techniques, including Harnack inequalities of Li-Yau-Hamilton type, the strong maximum principle for systems, dimension reduction, and the study of the geometry at infinity of noncompact stationary solutions (see§§1426 of [ H2].) In terms of Hamilton’s program, at least two obstacles remain: obtaining an injectivity radius estimate for Type II solutions and ruling out the so-called cigar soliton (the unique complete stationary solution on a surface with positive curvature) as the dimension reduction of a Type II singularity model. ∗Research partially supported by NSF grant DMS-9971891. 1In fact, Hamilton and Yau have announced informally that these are the only two obstacles and that they both would follow from obtaining a suitable differential matrix Harnack inequality of Li-Yau-Hamilton type for arbitrary solutions of the Ricci flow on compact 3manifolds. 2Added in proof: Very recently, Perelman [P1] has given a proof of the Little Loop Conjecture in all dimensions without curvature restriction. See also [P2] for some further developments.

Hamilton's maximum principle for systems states that given a reaction-diffusion equation (semi-linear heat-type equation) for sections of a vector bundle over a manifold, if the solution is initially in a subset invariant under parallel translation and convex in the fibers and if the ODE associated to the PDE preserves the subset, then the solution remains in the subset for positive time. We generalize this result to the case where the subsets are time-dependent and where there is an avoidance set from which the solution is disjoint. In applications the existence of an avoidance set can sometimes be used to prove the preservation of a subset of the vector bundle by the PDE.

The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the asymptotic scalar curvature ratio of a complete noncompact ancient Type I-like solution to the Ricci flow with bounded positive sectional curvature on an orientable 3-manifold, is infinity., Comment: 28 pages, no figures

We give a new and complete proof of Hamilton's injectivity radius estimate for sequences with bounded and almost nonnegative curvature operators, unbounded diameters, and bump-like origins. Such sequences arise in particular from dilations about a singularity of the Ricci flow on a 3-manifold.

We show that recent work of Ni and Wilking (in preparation) [11] yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kahler–Ricci shrinkers by Feldman, Ilmanen, and Knopf (2003) [7] exhibit that this result is sharp. We also prove a similar result for certain noncompact steady gradient Ricci solitons.

We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow.

We show that for a complete solution to theRicci–Kahler flow where the curvature, the potential andscalar curvature functions and their gradients are bounded depending ontime, the absolute value of both the scalar curvature and the gradientsquared of a modified potential function are bounded byC/t.

We consider the (degenerate) parabolic equationu t =G(▽▽u + ug, t) on then-sphereS n . This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, whereu is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate ¦▽u(·,t)¦

In the paper of Li and Yau [LY], a differential Harnack inequality was proved for the heat equation on a Riemannian manifold. Their technique was based upon the maximum principal which made possible the extension to geometric evolution equations. In particular, Richard Hamilton proved differential Harnack inequalities for the mean curvature flow [H1] and the Ricci flow [H2]. He also extended the result of Li-Yau and proved a matrix Harnack inequality for the heat equation [H3]. Recently, Ben Andrews [A] has proved differential Harnack inequalities for very general curvature flows of hypersurfaces, including anisotropic flows. The purpose of this paper is to give a geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. We shall show that the Harnack quantity is in fact the curvature of a torsion-free connection compatible with a degenerate metric on space-time. More precisely, let ( M, g( t )) be a solution to the Ricci flow