Scalar curvature

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 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, 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 global smooth Calabi flows on polarized Kähler surfaces $(M,L)$, assuming uniformly bounded Sobolev constants and the algebraic conditions $\chi ({\mathcal{O}}_M)=1$ and $c_1(M)\cdot c_1(L)>0$. Using the orbifold compactness theorem, we prove a uniform Bergman estimate, construct a normal test configuration for each geometric limit, and establish uniqueness of the extremal orbifold limit up to polarized holomorphic isometry. K-polystability then identifies the limit with the original surface and gives a constant scalar curvature Kähler metric. We also prove finite-group equivariant versions of these results. Independently of the flow, we obtain a uniform Bergman lower bound for polarized cscK surfaces with bounded volume and Sobolev constants and $|S|\le 1$. We illustrate the hypotheses with nonstationary flows near a product cscK metric on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and near Calabi's non-cscK extremal metric on the first Hirzebruch surface. Their controlled-cone energy inequalities follow from elementary calculations and continuity.

On every unit-volume flat complex three-torus, we construct a solution of the volume-preserving conformal Chern–Calabi flow with a finite-time Chern Type-I singularity at any prescribed point. The full spatial supremum has an exact asymptotic coefficient, and prescribed rescalings converge to a complete nonflat radial shrinking solution. The positive part of Chern scalar curvature can remain arbitrarily small throughout the flow, as can each prescribed subcritical spatial curvature norm. We determine the punctured terminal metric, its one-point completion and unique nonflat tangent cone, and the terminal energy asymptotics. We also exclude Chern Type-II under a logarithmically improved critical weak $L^3$ bound on Chern scalar curvature, and prove a separate scale-invariant integral criterion involving first Chern–Ricci curvature. The constructed singular solutions satisfy both criteria.

In this paper, we establish a new estimate on the time-derivative of pointed Nash entropy and use it to give a new proof of backward pseudolocality for Ricci flow. The proof does not rely on Bamler's compactness theorem and structure theory concerning limits of Ricci flows. Our methods also work for other reasonable geometric flows, like the $G_2$-Laplacian, at a scale depending on its deviation from Ricci flow and its scalar curvature.

In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local $\mu$-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the $G_2$-Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

Let $(M^n,g,f)$ be a complete connected shrinking gradient Ricci soliton with constant scalar curvature, satisfying $$\text{Ric}+{\nabla }^{2}f=\frac{1}{2}g.$$ We prove that if the focal set of $f$ is totally geodesic, then $M$ is rigid. Equivalently, its universal cover is a product of a compact Einstein manifold and a Euclidean space.

In this paper, we prove the long time existence of the pseudo-Calabi flow on a compact toric variety for any $T^n$-invariant initial Kähler metric. If the Mabuchi functional is proper modulo automorphisms, the flow converges exponentially to a constant scalar curvature Kähler (cscK) metric. In particular, this gives a flow proof that properness of the K-energy implies the existence of a cscK metric in the toric setting.

We study the volume-normalized gradient flow of the squared Chern scalar curvature on compact Hermitian manifolds of complex dimension $n\ge 3$. The flow extends smoothly past a finite time if the spatial supremum of Chern scalar curvature belongs to $L^2$ in time. A uniform curvature bound in the evolving $L^p$ norm, for one fixed $p>n$, gives global existence and smooth convergence of the whole trajectory to a constrained critical metric. On Chern scalar-flat backgrounds, small initial $L^p$ curvature gives exponential convergence to the background. A corresponding condition on the trace-free first Chern–Ricci form gives exponential convergence on Chern–Ricci-flat Hermitian and positive Kähler–Einstein backgrounds. The smallness thresholds are independent of the initial higher derivatives. We also construct constant Chern scalar curvature backgrounds of either sign with arbitrarily small smooth perturbations whose solutions, if global, cannot converge back to the background.

In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial $C^0$ estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

Let $X$ be a compact Kähler manifold with semi-ample canonical bundle. For any Kähler class $[\omega ]$, Jian–Shi–Song proved that there exists a unique cscK metric in the class $2\pi c_1(K_X)+\epsilon [\omega ]$ when $\epsilon \in (0,{\epsilon }_0]$, using the properness criterion of Chen–Cheng. We prove that the cscK metrics in $2\pi c_1(K_X)+\epsilon [\omega ]$ converge, as $\epsilon \downarrow 0$, in the Gromov–Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model of $X$ as conjectured by Jian–Shi–Song. Based on the technique developed in this paper, we can also give a new proof the theorem of Lee–Tosatti–Zhang on the Gromov-Hausdorff convergence of the normalized Kähler–Ricci flow on $X$.

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.

In this note, we prove that on a compact Hermitian manifold with negative Gauduchon degree, the Chern-Yamabe flow introduced in our recent work [13] converges to the solution of prescribed Chern scalar curvature problem, provided that the candidate curvature function is nonpositive and not identically zero.

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 study gradient Ricci solitons whose potential functions have level hypersurfaces with parallel second fundamental form. We show that such a soliton is locally a multiply warped product of a one-dimensional base and Einstein fibers, with potential function depending only on the base. Thus, locally, this condition characterizes the multiply warped structure that occurs in many classical constructions of Ricci solitons. We further prove that the same local structure follows for any gradient Ricci soliton if just one regular level hypersurface has parallel second fundamental form and parallel intrinsic Ricci tensor. As an application, we prove that a complete nonsteady gradient Ricci soliton with constant scalar curvature is rigid as soon as one regular level hypersurface has this property. In particular, the conjecture of Cao holds for gradient shrinking Ricci solitons with this property. We also prove that a complete gradient shrinking Ricci soliton whose regular level hypersurfaces have parallel second fundamental form and at most one nonzero principal curvature is rigid, without any assumption on the scalar curvature.

In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform $L^2$ Riemannian curvature bounds and $L^4$ Ricci curvature bounds up to dimension 7. These estimates improves $L^{2-\epsilon }$ curvature bounds established by Bamler and $L^2$ curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform $L^2$ curvature bounds on $n$-manifolds satisfying an integral $L^p$ Ricci curvature bound for $p>n/2$, a uniform volume lower bound, and a strong $\epsilon$-regularity assumption. This result extends the celebrated $L^2$ curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an $L^p$ adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori $L^{4-\epsilon }$ curvature radius bound of Ricci flow established by Bamler, we obtain uniform $L^2$ curvature bounds on Ricci flows with bounded scalar curvature of dimension $n\le 7$.

We prove a uniform scalar curvature bound, depending only on the dimension, for Kähler–Ricci shrinkers. We also prove that every noncompact Kähler–Ricci shrinker has a unique tangent cone at infinity, naturally homeomorphic to the base of its polarized Fano fibration. For shrinkers with maximal volume growth, this cone is isometric to the metric completion of the regular part equipped with the intrinsic distance induced by the limiting Kähler metric.

Let $G$ be a compact connected semisimple Lie group, let $Q$ be a bi-invariant metric and let $\rho$ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with $[H_0]=[\rho ]$ exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form $[H_0]=[\rho ]$ and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.