Sphere theorems

We prove that a complete connected four-dimensional gradient steady Ricci soliton with positive curvature operator that is $\kappa$-noncollapsed at all scales admits an effective isometric $O(3)$ action preserving the potential function and having principal orbits diffeomorphic to $S^2$.

We study quermassintegral-preserving curvature flows in ${\mathbb{H}}^{n+1}$, $n\ge 3$, driven by the curvature quotient $(E_k/E_{\ell }{)}^{1/(k-\ell )}$. For $1\le \ell <k$ with $k+\ell \le n$, we prove that positive sectional curvature is preserved. Starting from a smooth closed strictly convex hypersurface with positive sectional curvature, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. As an application, we establish sharp Alexandrov–Fenchel inequalities between $W_k$ and $W_{\ell }$ for smooth bounded convex domains whose boundaries have nonnegative sectional curvature, whenever $0\le \ell <k\le n$ and $k+\ell \le n$. Equality holds if and only if the domain is a geodesic ball. Together with previously known cases, these inequalities give the full family of Alexandrov–Fenchel inequalities for this class of domains in ${\mathbb{H}}^6$. We also construct counterexamples showing that $k+\ell \le n$ is the optimal range for preservation of positive sectional curvature by these flows.

We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

We extend the classification of manifolds with positive isotropic curvature to all dimensions $n\ge 5$. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact $\kappa$-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

We establish the eventual monotonicity of the Willmore deficit ${\int }_{\Sigma }{H}^{2}d\mu -16\pi$ along the mean curvature flow of smooth closed embedded strictly convex surfaces in ${\mathbb{R}}^3$, thereby complementing Huisken's exponential convergence with an eventual monotonicity property. As a consequence, the Hawking mass is eventually nondecreasing along the contracting mean curvature flow, in contrast with the usual inclusion monotonicity expected of the Hawking mass. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.

We study gradient Ricci solitons on partially totally real (PTR) submanifolds of complex space forms. In nonzero ambient holomorphic sectional curvature, the Gauss–soliton equation reconstructs the square $P^2$ of the tangential Kähler operator from the Hessian endomorphism and an extrinsic Ricci operator. Consequently, the PTR tangency spectrum is encoded by intrinsic soliton data together with the second fundamental form. For compact minimal PTR solitons with positive ambient curvature we obtain $\lambda \le c(n-1)/4$, with a strict version under a relative-nullity condition. We also prove compact triviality criteria, an exact trace-free anisotropy identity, a parallel-data rigidity result, and a nonexistence theorem for compact real hypersurfaces. A flat multi-level Gaussian model and a nonflat Segre $CR$-product provide explicit consistency checks and identify the limiting cases of the reconstruction formulas.

We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE $\frac{\mathrm{d}}{\mathrm{d}t}R=Q(R)$ in dimensions $n=7,8$, thereby extending the pinching estimate established by Brendle for $n\ge 12$ and by Chen for $9\le n\le 11$. In dimension $n=8$, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at $n=8$ by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension $n=7$, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension $n=8$. The $n=8$ pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible $(n-1)$-dimensional space forms, extending a theorem of Brendle from $n\ge 12$. Together with curvature-improvement and classification results of Cho–Li and Brendle–Naff, it also yields the classification of noncompact $\kappa$-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho–Li from $n=4$ or $n\ge 12$.

For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}{\mathbb{P}}^m\looparrowright {\overline{\mathbb{B}}}^{,N}(1)$ satisfies $\kappa (F{)}^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $\kappa (F)\le \sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}{\mathbb{P}}^3$, or $S^2\times S^1$. All three possibilities occur, while $\kappa (F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh–Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar–systolic inequality $$(\underset{Y}{\min }{R}_g)\text{sys}(g{)}^2<6{\pi }^2$$ for every spherical three-space form $Y$ with $|{\pi }_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar–systolic inequality for $\mathbb{R}{\mathbb{P}}^3$ of Bray–Brendle–Eichmair–Neves.

We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we characterise $3$-dimensional Ricci solitons of constant length and of finite volume on almost $\alpha$-cosymplectic manifolds. In fact, we show that these spaces are indeed compact quotients of Euclidean spaces. We also study complete, non-compact and finite volume contact metric manifolds in dimension $3$ admitting a Reeb vector field as a Ricci soliton. We show that it is a compact spherical space form. We include some examples to illustrate the necessity of the finite-volume and constant-length assumptions.

We obtain the unique asymptotics of $SO(k)\times SO(n-k+1)$-invariant, compact, simply-connected, factorwisely non-self-similar $n$-dimensional $\kappa$-solutions of the Ricci flow $(M^n,g(t))$, where $n\ge 4$ and $2\le k\le n-2$. More precisely, these $\kappa$-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere $S^n$, having a positive curvature operator metric $g(t)$ and a cylindrical tangent flow at $-\infty$, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric $g(t)$ of every $SO(k)\times SO(n-k+1)$-invariant ancient oval is represented in the form $g(t)=dz\otimes dz+F^2(z,t)g_{S^{k-1}}+G^2(z,t)g_{S^{n-k}}$ (up to flipping $k-1$ and $n-k$). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function $G(z,t)$, and prove that the uniqueness of $G(z,t)$ implies the uniqueness of $F(z,t)$. In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional $\kappa$-solutions of the Ricci flow.

Kong and Liu introduced the concept of hyperbolic Ricci flow in 2007 and used it to study the wave character of metrics. After that, many mathematicians have used this new geometric flow to study the evolution of manifolds and their structures. Ricci solitons and hyperbolic Ricci solitons are self-similar solitons of the Ricci flow and hyperbolic Ricci flow respectively. In this paper, we introduce the concept of hyperbolic $\ast -$Ricci solitons and hyperbolic Ricci-Yamabe solitons on a trans-Sasakian space forms and characterized the nature of some hyperbolic solitons. Additionally, we deduce the Ricci tensors of submanifolds of trans-Sasakian space forms and conformal trans-Sasakian space form and found the nature of solitons on submanifolds. Finally, we have included an example which will justify our result.

We prove that the Ricci flow preserves positive sectional curvature on homogeneous spheres and complex projective spaces. In conjunction with prior results, this completes the classification of which homogeneous spaces have positively curved metrics flowing outside the set of positively curved metrics and which do not.

We show that $\kappa$-solutions to the Ricci flow in dimensions $n\ge 4$ whose asymptotic shrinking Ricci soliton is the round cylinder ${\mathbb{S}}^{n-1}\times \mathbb{R}$ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.

We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space ${\mathbb{H}}^{n+1}$. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain conditions. This class of flows includes the $k$th mean curvature flow as a special case. We show that if the initial hypersurface has positive sectional curvature, then this property is preserved along the flow, and the evolving hypersurface contracts to a round point in finite time.

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.

In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].

We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p)\times O(q)$ symmetry in dimension $p+q$ for any pair of integers $p,q\ge 2$ and discuss their asymptotic geometry.

Lorantzian trans-Sasakian space form is a special type of space form in which the nature of even and odd dimensional space form both exist. Various curvature tensors with respect to Levi-Civita connection on the space form are derived in this paper. We have shown that if an odd-dimensional Lorentzian trans-Sasakian space form admits a hyperbolic Ricci soliton and hyperbolic conformal Ricci soliton then they will be $\eta$-Einstein. We also obtained the conditions for the solitons to be expanding, steady or shrinking. Finally, an example has been constructed which justifies the results obtained.

For all dimensions $n\ge 5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that ${\nabla }^{2}f$ is $2-$nonnegative and the curvature tensor is WPIC1 at some point $\bar{x}\in M$. Then $(M,g)$ must be a quotient of either $S^n$ or $S^{n-1}\times \mathbb{R}$. Our result partially extends the classification result for 4-dimensional PIC shrinking Ricci solitons established in [LNW16] to highter dimensions. Combining the pinching estimates deduced in [Chen24] we also extend the result in [CL23] to dimensions $n\ge 9$. Namely that a complete ancient solution to the Ricci flow of dimension $n\ge 9$ with uniformly PIC must be weakly PIC2.

This paper studies minimal surface entropy (the exponential asymptotic growth of the number of minimal surfaces up to a given value of area) for negatively curved metrics on hyperbolic $3$-manifolds of finite volume, particularly its comparison to the hyperbolic minimal surface entropy in terms of sectional and scalar curvature. On one hand, for metrics that are bilipschitz equivalent to the hyperbolic metric and have sectional curvature bounded above by $-1$ and uniformly bounded below, we show that the entropy achieves its minimum if and only if the metric is hyperbolic. On the other hand, by analyzing the convergence rate of the Ricci flow toward the hyperbolic metric, we prove that among all metrics with scalar curvature bounded below by $-6$ and with non-positive sectional curvature on the cusps, the entropy is maximized at the hyperbolic metric, provided that it is infinitesimally rigid. Furthermore, if the metrics are uniformly $C^0$-close to the hyperbolic metric and asymptotically cusped, then the entropy associated with the Lebesgue measure is uniquely maximized at the hyperbolic metric.