New papers on Ricci flow

In this paper, we first prove Appleton's conjecuture on the uniqueness of noncollapsed steady solitons constructed in. We also classify non-Ricci-flat $\kappa$-noncollapsed cohomogeneity one steady gradient Ricci solitons in dimension four and five. After passing to a finite cover, in dimension four, they must be the Bryant soliton or Appleton's noncollapsed 4d steady soliton; in dimension five, they must be the Bryant soliton or solitons constructed by Ivey–Dancer–Wang.

In this paper, we study the anisotropic mean curvature flow in the Gaussian space endowed with a Finsler metric obtained from a Minkowski metric by a conformal deformation. We first derive the conformal transformation formulas for the fundamental tensor, the Cartan tensor, and the Chern connection of the conformally related Finsler metrics. Based on these formulas, we express the anisotropic mean curvature in the Gaussian space in terms of the corresponding anisotropic mean curvature with respect to the original Minkowski metric, and derive the associated evolution equation. Finally, we establish the short-time existence and uniqueness of smooth solutions by reformulating the geometric evolution equation as a strictly parabolic quasilinear partial differential equation.

We prove that a complete noncompact gradient shrinking Ricci soliton of dimension at least two is a finite quotient of an Einstein cylinder if its scalar curvature is bounded below by the cylindrical value. The corresponding pointwise strict bound implies compactness. We also establish one-endedness criteria using curvature information at critical points of the potential, including degenerate ones, and obtain sharp one-endedness and rigidity results under global Ricci norm bounds.

We establish uniqueness of Ricci flows with non-negative sectional curvatures and scaling invariant curvature bounds, assuming that the induced distance functions converge to the same initial distance function up to isometry. In the compact case, restricting the size of the initial singular set, we also establish uniqueness under suitable weaker conditions.

We study the Gromov-Hausdorff stability of round spheres under perturbations of Perelman's nu-entropy. We show that in dimension two almost maximal entropy, together with an area normalisation, implies Gromov-Hausdorff closeness to the round two-sphere. In contrast, in dimensions at least three, we construct smooth metrics on the sphere whose entropies converge to that of the round sphere, but which remain a definite Gromov-Hausdorff distance away from it. These metrics can be chosen conformal and uniformly bi-Lipschitz equivalent to the round metric, with the same volume as the round sphere. Finally, in dimension three, we contrast this Gromov-Hausdorff instability with a dynamical stability result: metrics with nu-entropy above that of the round shrinking cylinder converge exponentially fast to round metrics under the volume-normalised Ricci flow.

In this paper, we investigate Ricci solitons in the para-Kenmotsu setting with respect to a $\phi$-related metallic semi-Riemannian metric arising from the underlying almost paracontact metric structure, under the assumption that the potential vector field is pointwise collinear with the Reeb vector field. We first establish the relation $\text{tS}=S+2n\beta (g-\eta \otimes \eta )$ between the Ricci tensors of the original metric $g$ and the $\phi$-related metallic metric $\mathrm{tg}$. We then show that the Ricci soliton equation implies that the original metric $g$ is of $\eta$-Einstein type. By combining this condition with the contracted Bianchi identity, we obtain a rigidity result implying $f=0$, $\lambda =\frac{2n}{C}$, $S=-2ng$, and consequently $\text{tS}=-\frac{2n}{C}\mathrm{tg}$. Thus, every Ricci soliton in this setting whose potential vector field is pointwise collinear with the Reeb vector field is necessarily trivial, and both $g$ and $\mathrm{tg}$ are Einstein. Finally, we provide two five-dimensional para-Kenmotsu examples: one realizing the resulting trivial Einstein Ricci soliton and another showing that failure of the required Einstein condition prevents the existence of such a soliton.

In every dimension $n\ge 3$, we show there exist Lagrangian mean curvature flows that develop neck pinch singularities such that the mean curvature remains uniformly bounded. This follows from a related result that, under a structural assumption on the singularity formation, the mean curvature at small scales near the singularity can be bounded by the Lagrangian angle at outer scales with suitable weights. Applying this result to a subset of the finite-time Lagrangian mean curvature flow singularities constructed in the companion paper [arXiv:2607.03152] and the infinite-time Lagrangian mean curvature flow singularities constructed by Tsai, Wood, and the second-named author [arXiv:2401.02228] then gives examples of both finite-time and infinite-time singularity formation with uniformly bounded mean curvature.

We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if $(M^n,g(t){)}_{t\in [0,T]}$ is a complete Ricci flow with nonnegative scalar curvature at $t=0$, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor $\mathrm{Ric}-\frac{1}{2}Rg$ is bounded above, then parabolicity is preserved along the flow and the conjugate radius of $g(t)$ is finite for every $t\in (0,T]$. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that ${\mathrm{liminf}}_{r\to \infty }\text{Vol}(B(p,r))/r^2=0$, has finite conjugate radius.

We prove that every smooth closed strictly convex hypersurface in ${\mathbb{R}}^{n+1}$ contracts to a round point under the ${\sigma }_k$-curvature flow for $2\le k\le n-1$, without additional pinching or symmetry assumptions. After rescaling about the extinction point to mean width two, the hypersurfaces converge smoothly and exponentially in normalized time to the unit sphere. The proof is based on an entropy which is invariant under translations and dilations and admits a representation by weighted boundary areas of orthogonal projections. A strengthened Codazzi inequality yields its monotonicity. An upper bound for the entropy controls the inradius relative to the mean width, while its dissipation controls the spatial variation of the normalized speed. We also obtain two-sided curvature bounds at the natural extinction scale.

In this article we produce many examples of knotted self expanders. For example, there is an expander isotopic to a trefoil knot with three half cylinders attached.

We prove that, in every dimension, the focal set of the potential function of a complete gradient shrinking Ricci soliton with constant scalar curvature is totally geodesic whenever it has dimension two or three, and that every complete five-dimensional gradient shrinking Ricci soliton with constant scalar curvature is rigid. The first result follows from a Bochner identity for the covariant derivative of the Ricci tensor at the focal set. For the second, when the scalar curvature is twice the soliton constant, we prove that total geodesy implies rigidity through an integral identity on the level hypersurfaces; the case of scalar curvature three times the soliton constant is treated by a separate argument. As further applications, we combine our methods with the rigidity theorem in the recent preprint of Li, Li, Ou, Qu, Wu and Zhou to obtain rigidity in arbitrary dimension when the scalar curvature is twice or three times the soliton constant, and under integral Weyl pinching for four-dimensional focal sets.

We study curvature flows for the dual Orlicz–Minkowski problem for strictly convex capillary hypersurfaces in the Euclidean half-space. The prescribed function has the form $\phi (h)G(X)$, where $X=\nabla h+h\nu$ and $G$ may be nonradial and nonhomogeneous. For acute contact angles, the evenness assumption together with either of the two integral conditions yields a smooth weighted-volume-preserving flow for all $t>0$ and a smooth solution to the dual Orlicz–Minkowski equation. Without the evenness assumption, two strict spherical-cap barriers yield the same conclusions for an unnormalized flow, whose limiting hypersurfaces solve the dual Orlicz–Minkowski equation with $c=1$. A uniform strict monotonicity condition under rescaling ensures uniqueness for $c=1$ and convergence of the unnormalized flow in any $C^k$ norm at an exponential rate. The problem considered here includes the capillary $L_p$-Minkowski and $L_p$ dual Minkowski problems as special cases.