Ricci flow papers from 2024

We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation builds on previous articles by Huisken and the third author, as we apply their techniques to establish new results in this geometric setting. Moreover, under some natural geometric assumptions, the noncompact case is also solved

Mayer asks a question what closed, embedded and nonconvex initial curves guarantee that Gage's area-preserving flow (GAPF) exists globally. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this paper, we prove this conjecture by using Dittberner's singularity analysis theory. A star-shaped "flying wing" curve is constructed to show that GAPF may not always preserve the star-shapedness of evolving curves. This example is also a negative answer to Mantegazza's open problem whether the curve shortening flow (CSF) always preserves the star shape of the evolving curves.

We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We prove that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. Further, we show that the resulting STCMC surfaces form an asymptotic foliation that is unique within the a-priori class.

This manuscript investigates the curvature and topological properties of certain $\infty$-Einstein Finsler metrics on Finsler metric measure spaces. By imposing symmetry conditions, we construct a series of special metrics and analyze their equivalence on special manifolds. Provided a Ricci curvature bound, we establish a linear growth lower bound estimate for the S-curvature and the distortion, revealing the interplay between curvature and measure on $\infty$-Einstein Finsler manifolds. Furthermore, by introducing scalar curvature and imposing a linear growth lower bound condition, we derive upper and lower bounds for the distortion, S-curvature, and the scalar curvature itself on asymmetric essential gradient Ricci solitons with certain non-Riemannian curvature constraints. These results yield direct topological finiteness conclusions for some forward-complete $\infty$-Einstein Finsler manifolds. Our work partially addresses Gromov's conjecture of scalar curvature in the context of Finsler metric measure spaces and provides a foundation for further research in geometric analysis within general Finsler geometry.

This paper investigates the volume-preserving harmonic mean curvature flow in asymptotically Schwarzschild spaces. We demonstrate the long-time existence and exponential convergence of this flow with a coordinate sphere of large radius serving as the initial surface in the asymptotically flat end, which eventually converges to a constant harmonic mean curvature surface. We also establish that these surfaces form a foliation of the space outside a large ball. Finally, we utilize this foliation to define the center of mass, proving that it agrees with the center of mass defined by the ADM formulation of the initial data set.

We construct a continuous 3-parameter family of non-shrinking Ricci solitons complex line bundles $O(k)$ over ${\mathbb{CP}}^{2m+1}$, where the base space is not necessarily Kähler–Einstein. Each $O(k)$ with $k\in [3,2m+1]$ admits at least one asymptotically conical (AC) Ricci-flat metric in this family. For each $O(k)$ with $k\ge 3$, the family includes infinitely many asymptotically paraboloidal (AP) steady Ricci soliton.

We show the existence of a properly immersed translating solution to curve diffusion flow in the plane. Curve diffusion flow is a higher order version of curve shortening flow, namely $${\left(\frac{dX}{dt}\right)}^{\perp }=-{\kappa }_{ss}N.$$

In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case.

In this paper, we study $K^{\alpha }$–translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space ${\mathbb{E}}^3$. First, we investigate the condition under which two parallel surfaces can become $K^{\alpha }$–translators moving with the same speed $w$. Then, we examine $K^{\alpha }$–translators on canal surfaces and we show that if a canal surface is $K^{\alpha }$–translator, then it must be a surface of revolution in ${\mathbb{E}}^3$. We also provide examples for moving a surface of revolution under $K$–flow (Gauss curvature flow) and $K^{-1/2}$–flow (inverse Gauss curvature flow) along a direction $w=(0,0,1)$ and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no $K^{\alpha }$–translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed $w$, while the such rotational surfaces itself is a $K^{\alpha }$–translator with speed $w$.

Recent interest among geometers in $f$-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by V. Rovenski and R. Wolak as a generalization of Hermitian and Kähler structures, as well as $f$-structures, allow a fresh look at the classical theory. In this paper, we study a new $f$-structure of this kind, called the weak $\beta$-Kenmotsu $f$-structure, as a generalization of K. Kenmotsu's concept. We prove that a weak $\beta$-Kenmotsu $f$-manifold is locally a twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature.

We would like to study new Ricci flow invariant curvature conditions. Specifically, we provide quantitative evidence for an unpublished conjecture of Böhm and Wilking. As an application, we study the topology of manifolds with pinched curvature.

This article introduces the $L_p$-Gauss dual curvature measure and proposes its related $L_p$-Gauss dual Minkowski problem as: for $p,q\in \mathbb{R}$, under what necessary and/or sufficient condition on a non-zero finite Borel measure $\mu$ on unit sphere does there exist a convex body $K$ such that $\mu$ is the $L_p$ Gauss dual curvature measure? If $K$ exists, to what extent is it unique? This problem amounts to solving a class of Monge-Ampère type equations on unit sphere in smooth case: \beginalign e^{-\frac{|\nabla h_K|^2+h_K^2}2}h_K^1-p (|\nabla h_K|^2+h_K^2)^{\fracq-n2} \det(\nabla^2h_K+h_KI)=f,\qquad (0.1) \endalign where $f$ is a given positive smooth function on unit sphere, $h_k$ is the support function of convex body $K$, $\nabla h_K$ and ${\nabla }^2{h}_K$ are the gradient and Hessian of $h_K$ on unit sphere with respect to an orthonormal basis, and $I$ is the identity matrix. We confirm the existence of solution to the new problem with $p,q>0$ and the existence of smooth solution to the equation (0.1) with $p,q\in \mathbb{R}$ by variational method and Gaussian curvature flow method, respectively. Furthermore, the uniqueness of solution to the equation (0.1) in the case $p,q\in \mathbb{R}$ with $q<p$ is established.