Curvature pinching

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.

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 derive an interior planarity estimate for space-time proper mean curvature flows of $n$-dimensional submanifolds of ${\mathbb{R}}^{n+p}$, $n\ge 5$, that are uniformly quadratically pinched up to the Andrews-Baker threshold $\frac{4}{3n}$. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for $n\ge 5$, and by known results for $2\le n\le 4$, every space-time proper ancient solution that is uniformly $\frac{4}{3n}$-pinched is either a static affine $n$-plane or lies in a fixed affine $(n+1)$-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.

We develop a pinching method for mean curvature flow in higher codimension. Motivated by pinching constructions for Ricci flow, the method uses a nonlinear deformation of the second fundamental form. We introduce two tensorial curvature conditions for closed immersions in Euclidean space which extend beyond the scope of classical scalar pinching results. In dimensions at least ten, we apply the method to prove convergence to a round point under the first condition and a canonical neighborhood theorem under the second.

We establish local convexity and cylindrical pinching estimates for hypersurface mean curvature flow under an intermediate convexity condition and integral bounds on the mean curvature. These estimates hold after a positive waiting time and do not require pointwise upper bounds for the mean curvature on the parabolic boundary. In arbitrary codimension, we also establish local planarity estimates for properly defined flows satisfying a uniform $c_n$-pinching condition, without assuming bounded curvature. As applications, the hypersurface estimates yield convexity and cylindrical pinching for complete, spacetime-proper ancient hypersurface flows with uniform Euclidean area growth, without any bounded-curvature assumption. The planarity estimates imply that every complete, connected, spacetime-proper ancient flow satisfying the uniform $c_n$-pinching condition is either a static affine $n$-plane or lies in a fixed affine $(n+1)$-plane for all time.

In this paper, we study heat kernel estimates along the Kähler-Ricci flow with finite time noncollapsing singularities over a compact Kähler manifold. The starting point is the type-II scalar-curvature estimate, which gives the natural parabolic scale $r\lesssim T-t$ near the singular time. By combining the distance distortion and space-time cut-off estimates of Bamler–Zhang with entropy, mean-value inequality, and conjugate heat-kernel estimates, we obtain two-sided Gaussian bounds and gradient bounds that are uniform on this scale. Under the global type-I scalar-curvature estimate, the same estimates hold on every scale. We then use these estimates and the monotonicity of the $W_1$-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances $d_t$ (induced by $\omega (t)$) converge uniformly to this pseudo-distance. Its metric quotient is the unique Gromov–Hausdorff limit. Finally, we apply Bamler's theory and the recent work of Hallgren–Zhang to obtain the following. Every fixed-base-point type-I blow-up admits a subsequential metric-flow limit whose time slices are singular shrinking Kähler–Ricci soliton spaces with normal complex-analytic and log-terminal structure, unique tangent cones, and a polarized Fano fibration. For moving base points, the limiting metric flow, equipped with its limiting conjugate heat flow, is a metric soliton precisely under an entropy-pinching condition.

Let $(M^{2m},g,f,J)$ be a complete nonsteady gradient Kähler-Ricci soliton satisfying $\mathrm{Ric}+{\nabla }^{2}f=\lambda g$, $\lambda \ne 0$. We prove that constant scalar curvature forces the soliton to be rigid. More precisely, the universal cover splits holomorphically and isometrically as $N^{2k}\times {\mathbb{C}}^{m-k}$, where $N^{2k}$ is Kähler-Einstein with ${\mathrm{Ric}}_{g_N}=\lambda g_N$. In the shrinking case the quotient is trivial; in the normalization $\lambda =1/2$ one has $R\equiv k$. The Kähler result rests on a Riemannian rigidity criterion. On a complete nonsteady gradient Ricci soliton, if ${\mathcal{L}}_{\nabla f}\mathrm{Ric}$ is nonnegative or nonpositive everywhere, then it vanishes and the soliton is rigid; no assumption on the scalar curvature is needed. Along the Ricci flow generated by the soliton, this means that a Ricci tensor that is monotone in time is constant in time, and that this forces rigidity. We also give a direct proof that the pinching $0\le \mathrm{Ric}\le \lambda g$ forces constant scalar curvature and radial flatness, yielding the rigidity conclusion through the Petersen-Wylie characterization. The proof combines a weighted cutoff argument with a partial Codazzi symmetry for the Ricci endomorphism.

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 study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete $n$-dimensional connected translator isometrically immersed in $({\mathbb{R}}^{n+p},⟨\cdot ,\cdot ⟩)$ satisfies $$|B{|}^2\le \left(\frac{4}{3n}-{\epsilon }_0\right)|H{|}^2$$ for some ${\epsilon }_0>0$, then either it is an affine $n$-plane or $|H|>0$ everywhere and the translator is contained in an $(n+1)$-dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition $$|B{|}^2\le (c_n-{\epsilon }_0)|H{|}^2,c_n=\min \left\{\frac{4}{3n},\frac{1}{n-2}\right\}$$ for $n\ge 3$, with $c_n=2/3$ when $n=2$, we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an $(n+1)$-dimensional affine subspace. No entropy assumption is required.

Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least $7$, while in dimensions $5$ and $6$ a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions $5$ and $6$ throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.

Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander $x:M\to {\mathbb{R}}^{n+p}$($n\ge 3$) that satisfies both $({\int }_M|A{|}^{n}d\mu {)}^{\frac{1}{n}}<K(n)$ and ${\int }_M|A{|}^n{e}^{\frac{|x{|}^2}{2}}d\mu <\infty$ for a positive constant $K(n)$ depending only o isometric to ${\mathbb{R}}^n$. Moreover, we show that the rigidity result persists when the pinching condition is expressed in terms of the trace-free second fundamental form.

We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

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 $n\ge 12$.

Let $n\ge 3$ and $0\le \tau \le 2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint $n=3$, $\tau =2$, positive $\tau$-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.

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.

We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential $C^{\infty }$ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

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.

In this paper, we prove a spectral upper-pinching theorem for complete properly immersed self-shrinking hypersurfaces. Our argument is inspired by the second author's recent work. If ${\lambda }_{\rho }(\Sigma )\ge \lambda >0$ and $S=|A{|}^2<1+\lambda$, then $\Sigma$ is either a hyperplane or a generalized round cylinder. In the properly embedded case, the Ding–Xin and Brendle–Tsiamis weighted Poincaré estimate gives ${\lambda }_{\rho }(\Sigma )\ge 1/2$. Consequently, the pointwise upper pinching $S<3/2$ forces $\Sigma$ to be a hyperplane or a generalized round cylinder. For embedded self-shrinking surfaces in ${\mathbb{R}}^3$, we also obtain the endpoint case $S\le 3/2$. These results remove the lower pointwise pinching assumption in the corresponding embedded upper-pinching range and improve the ranges in earlier work of Ding–Xin , Cheng–Wei , and Lei–Xu–Xu .

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.