Pseudolocality of G₂-Laplacian Flow

Wed, 28 Oct 2026 00:00:00 GMT

Weiyong He, Longhuan Yan — 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 μ-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the G₂-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.

An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow

Tue, 27 Oct 2026 00:00:00 GMT

Hongxin Guo, Guozhi Lin — For a rotationally symmetric Ricci flow g(t)=ds²+f(s,t)²g_(Sⁿ⁻¹), n≥3, we study the scale-invariant curvature anisotropy A=1-fₛ²+ffₛₛ introduced by Angenent–Knopf. At the critical power p_*=(n-1)/(n-2), with Φ=sgn(A)|A|^(p_*/2), we prove the distributional identity (∂ₜ−Δ)|A|^(p_*) =−4/n-1 |Φₛ+(n-1)(fₛ)/fΦ|². Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on Sⁿ: if the entropy associated with a singular-time kernel is at most 1 at an earlier regular time, every tangent flow at the corresponding singular point is a round shrinking sphere. This criterion follows from the 0/1 quantization of the singular-time anisotropic density.

Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

Thu, 15 Oct 2026 00:00:00 GMT

Lei Zhang, Zhenlei Zhang — 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≲ 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 W₁-distance between conjugate heat flows to construct a canonical singular-time pseudo-distance. The distances dₜ (induced by ω(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.

Heat kernel on Ricci shrinker metric measure spaces

Wed, 16 Sep 2026 00:00:00 GMT

Bing Wang, Jie Wang — As a metric measure space possessing positive Bakry-Émery curvature, a Ricci shrinker with potential function f admits a heat kernel H_f under the weighted volume measure e^(−f)dv. In this paper, we study H_f systematically and establish a series of fundamental estimates. Moreover, we clarify the equivalence between H_f and the spacetime heat kernel H(x,t;y,s) under Ricci flow induced by a Ricci shrinker. Inspired by this relation, we extend corresponding analysis tools of Ricci flows to Ricci shrinker metric measure spaces, such as Nash entropy and reduced distance. As a direct application, we prove that |∇R|=o(f^(3/2)) implies R=o(f).

The Fisher Metric of the Ricci Flow Heat Kernel

Fri, 14 Aug 2026 00:00:00 GMT

Bennett Chow, Robert Koirala — We introduce and study a Fisher information metric g^F_τ associated to the conjugate heat kernel of a Ricci flow (Mⁿ,gₜ). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that g^F_τ is monotone in scale and satisfies g^F_τ≤ gₜ. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect gₜ−g^F_τ. This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities 0

Second-Order Departure of the Gigli–Mantegazza Flow from Ricci Flow

Fri, 14 Aug 2026 00:00:00 GMT

Dongwoo Gang — For a closed connected Riemannian manifold (M,g), the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding x↦ pₜ(x,·) dvol_g. The resulting family gₜ agrees with Ricci flow to first order in t, but in general not to second order. We prove that gₜ =g-2tRic_g +t²(−ΔRic_g +2Ric_g²−2/3𝓠_g) +O_(C⁰)(t³), where 𝓠_g is quadratic in the full curvature tensor. The term −ΔRic_g also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while gₜ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is O(t²), and round spheres show that this estimate is sharp.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows in dimension three

Thu, 11 Jun 2026 00:00:00 GMT

Yu Li — Starting from a closed, orientable three-dimensional Riemannian manifold, we consider the completion of the associated singular Ricci flow with respect to a natural spacetime distance. We show that this completion admits canonical intrinsic metrics on its time-slices, defined by conjugate heat kernel measures, and that these time-slices arise as metric limits of the regular part of the flow. More precisely, for any t₀>0 and any connected component Z_(t₀)' of the time-slice Z_(t₀) of the completion, we prove that (𝓡ₜ',d_(gₜ)) xrightarrow[t↗ t₀]mathrmGromov-Hausdorff (Z_(t₀)',d_(t₀)^Z), where 𝓡ₜ' is the corresponding connected component of the regular part and 𝓡ₜ' denotes its metric completion. In particular, this yields the Gromov-Hausdorff convergence at the first singular time for closed three-dimensional Ricci flows. We also establish a refined structure theory for the singular set of the completion. In particular, the singular set is horizontally parabolic 1-rectifiable, and its time image has vanishing 1/2-dimensional Hausdorff measure. Moreover, on each time-slice, the singular set has Minkowski dimension at most 1. The proof relies on heat kernel estimates for singular Ricci flows and the generalization of the structure theory of noncollapsed Ricci flow limit spaces.

Sharp Gaussian Isoperimetry along a Ricci Flow

Wed, 20 May 2026 00:00:00 GMT

Robert Koirala — We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein–Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein–Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's L^p-Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's Hₙ-centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.

The Ricci-DeTurck flow on complete manifolds

Tue, 24 Mar 2026 00:00:00 GMT

Jing-Bin Cai, Bing Wang — Based on the framework of Koch-Lamm and tensor heat kernel estimates, we obtain a uniform proof of the short-time existence, uniqueness, and continuous dependence for Ricci flows starting from a complete Riemannian metric with bounded curvature. A new ingredient is an effective continuous dependence estimate without the assumption of injectivity radius lower bound.

Well-posedness of mean curvature flow

Tue, 24 Mar 2026 00:00:00 GMT

Yongheng Han — In this paper, using heat kernel estimates and contraction mapping principle, we give a new proof of the existence and uniqueness of mean curvature flow starting from hypersurface with bounded second fundamental form. Moreover, we show the continuous dependence of mean curvature flow on initial data.

Dynamical functionals on ancient ARF Ricci flows

Thu, 13 Nov 2025 00:00:00 GMT

Isaac M. Lopez, Rio Schillmoeller — We introduce a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows with modest decay using limits of conjugate heat flows. This functional satisfies a steady Ricci breather-type rigidity and provides an upper bound for the ordinary λ-functional while retaining many of its properties. In addition, motivated by work of Colding and Minicozzi, we derive local eigenvalue estimates for normalized Ricci flows coupled with conjugate heat flows.

Well-posedness of Ricci Flow in Lorentzian Spacetime and its Entropy Formula

Mon, 22 Sep 2025 00:00:00 GMT

M. J. Luo — This paper attempts to construct monotonic entropy functionals for four-dimensional Lorentzian spacetime under physical boundary conditions, as an extension of Perelman's monotonic entropy functionals constructed for three-dimensional compact Riemannian manifolds. The monotonicity of these entropy functionals is utilized to prove the well-posedness of applying Ricci flow to four-dimensional Lorentzian spacetime for a long flow-time, particularly for the timelike modes which would seem blow up and ill-defined. The general idea is that the Ricci flow of a Lorentzian spacetime metric and the coupled conjugate heat flow of a density on the Lorentzian spacetime as a whole turns out to be the gradient flows of the monotonic functionals for a long flow-time, so the superficial "blow-up" in the individual Ricci flow system or the conjugate heat flow system contradicts the boundedness of the monotonic functionals within finite flow interval, which gives a semi-global control to the whole coupled system. The physical significance and applications of these monotonic entropy functionals in real gravitational systems are also discussed.

Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

Tue, 12 Aug 2025 00:00:00 GMT

Fabius Krämer — The famous thresholding scheme by Merriman, Bence, and Osher (Motion of multiple junctions: A level set approach. Journal of Computational Physics 112.2 (1994): 334-363.) proved itself as a very efficient time discretization of mean curvature flow. The present paper studies a multiphase, volume constrained version on a weighted manifold that naturally arises in the context of data science. The main result of this work proves convergence to multiphase, weighted, volume-preserving mean curvature flow on a smooth, closed manifold. The proof is only conditional in the sense that convergence of the approximating energy to the weighted perimeter is assumed. These type of assumptions are natural and common in the literature. The proof shows convergence in the energy dissipation inequality, induced by the underlying gradient flow structure, similar to the work of Laux and Otto (The thresholding scheme for mean curvature flow and De Giorgi's ideas for minimizing movements. The Role of Metrics in the Theory of Partial Differential Equations 85 (2020): 63-94.). This leads to a new De Giorgi solution concept that describes weakly the limiting flow. Estimates regarding the derivatives of the heat kernel are developed in order to pass to the limit in the variations of the energy and metric.

Perelman's entropy and heat kernel bounds on RCD spaces

Tue, 04 Mar 2025 00:00:00 GMT

Camillo Brena — We study Perelman's W-entropy functional on finite-dimensional RCD spaces, a synthetic generalization of spaces with Bakry-Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the W-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.

Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

Thu, 20 Jul 2023 00:00:00 GMT

Xiaolong Li, Hao-Yue Liu, Xin-An Ren — We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Kähler-Ricci flow. As an application, we obtain a monotonicity formula.

A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

Sat, 08 Jul 2023 00:00:00 GMT

Xingyu Song, Ling Wu, Meng Zhu — Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral iteration argument which utilizes Hamilton's gradient estimate, heat kernel Gaussian bounds and parabolic Harnack inequality. In this paper, we show that the sharp Li-Yau estimate can actually be obtained directly following the classical maximum principle argument, which simplifies the proof in [26]. In addition, we apply the same idea to the heat and conjugate heat equations under the Ricci flow and prove some Li-Yau type estimates with optimal coefficients.

Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

Fri, 16 Jun 2023 00:00:00 GMT

Xiaolong Li, Qi S. Zhang — In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.

Heat kernel on Ricci shrinkers (II)

Fri, 20 Jan 2023 00:00:00 GMT

Yu Li, Bing Wang — This paper is the sequel to our study of heat kernels on Ricci shrinkers in. In this paper, we improve many estimates in and extend the recent progress of Bamler. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of IF-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.

Nash Entropy and Heat Kernel on Generalized Ricci Flow

Sun, 11 Dec 2022 00:00:00 GMT

Xilun Li — We introduce analogous geometric quantities and prove some geometric and analytic bounds in [Bam20a] to generalized Ricci flow.

Scalar curvature, entropy, and generalized Ricci flow

Tue, 26 Jul 2022 00:00:00 GMT

Jeffrey Streets — We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction-diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles.

Tangent Flows of Kähler Metric Flows

Sun, 13 Feb 2022 00:00:00 GMT

Max Hallgren, Wangjian Jian — We improve the description of F-limits of noncollapsed Ricci flows in the Kähler setting. In particular, the singular strata 𝓢^k of such metric flows satisfy 𝓢^(2j)=𝓢^(2j+1). We also prove an analogous result for quantitative strata, and show that any tangent flow admits a nontrivial one-parameter action by isometries, which is locally free on the cone link in the static case. The main results are established using parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points, which may be of independent interest.

Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

Thu, 27 Jan 2022 00:00:00 GMT

Keita Kunikawa, Yohei Sakurai — Bamler-Zhang have developed geometric analysis on Ricci flow with scalar curvature bound. The aim of this paper is to extend their work to various geometric flows. We generalize some of their results to super Ricci flow whose Muller quantity is non-negative, and obtain Gaussian heat kernel estimates.

Hessian estimates for the conjugate heat equation coupled with the Ricci flow

Tue, 10 Aug 2021 00:00:00 GMT

Hong Huang — In this short note we obtain some local and global upper bounds for the Hessian of a positive solution to the conjugate heat equation coupled with the Ricci flow.

Ricci limit flows and weak solutions

Fri, 06 Aug 2021 00:00:00 GMT

Beomjun Choi, Robert Haslhofer — In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler's precompactness theorem, as well as every singular Ricci flow from Kleiner-Lott, is a weak solution in the sense of Haslhofer-Naber. We also generalize all path-space estimates from Haslhofer-Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.

Neckpinch — Heat kernel