Ricci flow papers from 2022

In the present work we find the Lie point symmetries of the Ricci flow on an $n$-dimensional manifold. and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations – seen as a "static" Ricci flow – , and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.

In this second part of a sequence of two papers, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry-Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the first paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, hypercubes) and exhibit further properties of this flow. One particular aspect in our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the available Python functions and routines available in the ancillary file. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

Space curve motion describes dynamics of material defects or interfaces, can be found in image processing or vortex dynamics. This article analyses some properties of space curves evolved by the curve shortening flow. In contrast to the classical case of shrinking planar curves, space curves do not obey the Avoidance principle in general. They can lose their convexity or develop non-circular singularities even if they are simple. In the first part of the text, we show that even though the convexity of space curves is not preserved during the motion, their orthogonal projections remain convex. In the second part, the Avoidance principle for spherical curves under the curve shortening flow in ${\mathbb{R}}^3$ is shown by generalizing the arguments developed by Hamilton and Gage.

We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.

We prove that the Riemannian Penrose Inequality holds for Asymptotically Flat $3$-manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the $\mathrm{ADM}$ mass being a well-defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio and the third named author. As a consequence, we establish the equality between $\mathrm{ADM}$ mass and Huisken's Isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose Inequality in terms of the Isoperimetric mass on any $3$-manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well-posed notion of weak Inverse Mean Curvature Flow. In particular, such Isoperimetric Riemannian Penrose Inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's Isoperimetric mass and the Hawking mass.

We prove a local gap theorem for Ricci shrinkers, which states that if the local $\mu$-functional at scale $1$ on a large ball centered at the minimum point of the potential function is close enough to $0$, then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global $\mu$-functional to be close enough to $0$. Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].

We introduce analogous geometric quantities and prove some geometric and analytic bounds in [Bam20a] to generalized Ricci flow.

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on ${\mathbb{S}}^2\times {\mathbb{R}}^2$ (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.

In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus-Sturzenhecker to such flows, the latter holding in low dimension and conditionally to a convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows to simplify many arguments.

Any expanding homogeneous Ricci soliton (in particular any homogeneous Einstein manifold of negative scalar curvature) can be obtained, up to isometry, from a Lie subgroup of a nilpotent Iwasawa group $N$ whose induced metric is a Ricci soliton. By nilpotent Iwasawa group we mean the nilpotent Lie group $N$ of the Iwasawa decomposition associated with a symmetric space of non-compact type. Motivated by this fact, in this paper we classify codimension one Lie subgroups of any nilpotent Iwasawa group $N$ whose induced metric is a Ricci soliton.

We introduce a new curvature-pinching condition, which is weaker than the positive sectional curvature or PIC1, and then we prove several rigidity results for the rotationally symmetric solutions of steady Ricci solitons, i.e., the Bryant Ricci solitons.