Ricci flow papers from 2021

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins-Sekerka flow.

We find examples of cohomogeneity one metrics on $S^4$ and $\mathbb{C}{P}^2$ with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

In a Riemannian manifold, the Ricci flow is a partial differential equation for evolving the metric to become more regular. We hope that topological structures from such metrics may be used to assist in the tasks of machine learning. However, this part of the work is still missing. In this paper, we propose Ricci flow assisted Eucl2Hyp2Eucl neural networks that bridge this gap between the Ricci flow and deep neural networks by mapping neural manifolds from the Euclidean space to the dynamically stable Poincaré ball and then back to the Euclidean space. As a result, we prove that, if initial metrics have an $L^2$-norm perturbation which deviates from the Hyperbolic metric on the Poincaré ball, the scaled Ricci-DeTurck flow of such metrics smoothly and exponentially converges to the Hyperbolic metric. Specifically, the role of the Ricci flow is to serve as naturally evolving to the stable Poincaré ball. For such dynamically stable neural manifolds under the Ricci flow, the convergence of neural networks embedded with such manifolds is not susceptible to perturbations. And we show that Ricci flow assisted Eucl2Hyp2Eucl neural networks outperform with their all Euclidean counterparts on image classification tasks.

We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles $\alpha \in (0,\pi )$. First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess.

The mathematical model of grain boundary motion, including lattice misorientations' effect, is considered. When time-dependent lattice misorientations are state variables of the surface tension of the grain boundary, to ensure the energy dissipation law, one can obtain a curvature flow of networks with time-dependent mobilities. This paper studies the solvability and long-time asymptotic behavior of the curvature flow subjected to the Herring condition which ensures that the constituent grain boundary surface tensions are balanced at the triple junction.

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\le cR$ for some positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for $4$-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a $3$-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Chodosh 14 in dimension three and show that any $3$-dimensional gradient Ricci expander $C^2$ asymptotic to $\left(C({\mathbb{S}}^2),d{t}^2+\alpha t^2{g}_{{\mathbb{S}}^2}\right)$ is rotationally symmetric, where $\alpha \in (0,1]$ is a constant and $g_{{\mathbb{S}}^2}$ is the standard metric on ${\mathbb{S}}^2$ with constant curvature $1$.

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to address these questions, partial results of Boehm, Lafuente, and Lauret are extended to show that left-invariant Ricci solitons on solvable Lie groups are unique up to scaling and isometry.

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

A point source of light is placed inside an oval. The $n$-th caustic by reflection is the envelope of the light rays emanating from the light source after $n$ reflections off the curve. We show that each of these caustics, for a generic point light source, has at least 4 cusps. This is a billiard variation on Jacobi's Last Geometric Statement, concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

In this paper, we prove a pseudolocality-type theorem for $\mathcal{L}$-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the $\mathcal{L}$-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L. Chen.

Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $\sigma$-models. We include non-conformal $\sigma$-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General $\sigma$-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry. This thesis is based on the conjecture that integrable $\sigma$-models are renormalizable, or stable under the RG flow. It is widely understood that classically integrable theories are stable at the leading 1-loop order with only a few parameters running. Here we address what happens at higher-loop orders. We find that integrable $\sigma$-models generally remain RG-stable at higher-loops provided they receive a particular choice of finite counterterms, or quantum (${\alpha }^{\prime }$) corrections to the target-space geometry. We explicitly construct these quantum corrections for examples of integrable $\eta$- and $\lambda$-deformed $\sigma$-models. We then reformulate the $\lambda$-models as $\sigma$-models on a "tripled" $G\times G\times G$ configuration space, where they become automatically renormalizable due to manifest symmetries and a decoupling of some fields. We also consider the integrable $G\times G$ and $G\times G/H$ models and construct a new class of integrable $G\times G/H$ models with abelian $H$. We then present a new and different link between integrability and the RG flow in the context of $\sigma$-models with 'local couplings' depending explicitly on 2d time. Such models are naturally obtained in the light-cone gauge in string theory, pointing to the possibility of a large, new class of solvable string models.

In this work we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a $1+1+2$ spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor with respect to the induced metric. These are slices of constant time in the spacetime. Firstly, the form of the Ricci tensor for general hypersurfaces is obtained and the conditions under which the general case reduces to those of constant time slices are specified. We provide a characterization of these hypersurfaces, with key physical quantities in the spacetime playing a role in specifying the local geometry of these hypersurfaces. Furthermore, we investigate the case where these hypersurfaces admit a Ricci soliton structure. The particular cases where the vector fields associated to the solitons are Killing or conformal Killing vector fields are analyzed. Finally, in the context of spacetimes with local rotational symmetry it is shown that, only spacetimes in this class with vanishing rotation and spatial twist can admit the hypersurface types considered, and that the hypersurfaces are necessarily flat. And if such hypersurface do admit a Ricci soliton structure, the soliton is steady, with the components of the soliton field being constants.