Curvature pinching

We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3,-\frac{3}{2}]$. The upper bound $-\frac{3}{2}$ is achieved by the complex Heisenberg group ${\mathrm{Heis}}_3(\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of ${\mathrm{Heis}}_3(\mathbb{R})\times {\mathbb{R}}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to ${\mathrm{Heis}}_3(\mathbb{R})\times {\mathbb{R}}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant $-\frac{3}{2}$ is also given, and the pinching constants of various examples are computed.

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 $n\ge 12$.

We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\ell =\lfloor \frac{n}{2}\rfloor$. We prove the sharp pointwise estimate $$q_2(E)\ge -\frac{2(\ell -1)}{3\ell }\mathrm{Scal}(E){\mathrm{Id}}_{{\Lambda }^2{V}^{\ast }}$$ for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying this estimate to the decomposition ${\mathrm{Rm}}_g=K_{\min }I+E$, we obtain the vanishing of $H^2(M;\mathbb{R})$ under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. Apart from the flat case, this yields $b_2(M)=0$ in odd dimensions and $b_2(M)\le 1$ in even dimensions. At even-dimensional endpoint, $b_2(M)>0$ forces $(M,g)$ to be isometric, up to scaling, to ${\mathbb{CP}}^{\ell }$ with its Fubini-Study metric. As a consequence every closed five-dimensional manifold satisfying the strict pinching condition implies is a rational homology sphere. An anisotropic rescaling of the same homogeneous four-frame estimate also gives the sharp pointwise sectional-scalar pinching criterion $$K_{\min }\ge \frac{n(n-1)}{n^2-n+12}{S}_0⟹PIC2.$$ The strict pinching places the curvature tensor in the interior of PIC2 and normalized Ricci flow brings it to a positive constant sectional curvature.

The stability and deformation theory of Einstein metrics traditionally relies on the classical Berger-Ebin transverse-traceless gauge, which structurally decouples the scalar trace from the divergence-free component of metric perturbations. In the present paper, we introduce a new spectral-geometric framework based on the Chen-Nagano gauge condition. This condition naturally arises from the harmonicity of the identity map and is intrinsically satisfied by the Ricci tensor itself via the contracted second Bianchi identity. Unlike the classical transverse-traceless framework, the Chen-Nagano gauge preserves a nontrivial interaction between the trace and trace-free sectors of a deformation. We establish a first-order differential relation proving that the divergence of the trace-free part is completely governed by the gradient of the scalar trace. Utilizing commutation formulas on Einstein manifolds, we derive a second-order spectral coupling relation that links the Lichnerowicz Laplacian to a shifted scalar operator. As a primary geometric consequence, we prove that under suitable spectral pinching assumptions, the Chen-Nagano gauge collapses to the classical transverse-traceless gauge. Specifically, we show that on compact connected negatively curved Einstein manifolds, any volume-preserving Chen-Nagano harmonic deformation whose trace-free component lies below a specific spectral threshold determined by the Einstein constant is necessarily transverse-traceless. Furthermore, we connect this rigidity to the curvature operator of the second kind, establishing explicit lower spectral bounds. Finally, we provide a dynamical interpretation within the Ricci flow framework, demonstrating that the linearized Ricci flow under the Chen-Nagano gauge reduces to a strictly parabolic equation governed by the Lichnerowicz Laplacian, ensuring exponential decay of admissible perturbations.

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 investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.

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.