Heat kernel

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.

For a rotationally symmetric Ricci flow $g(t)=d{s}^2+f(s,t{)}^2{g}_{{\mathbb{S}}^{n-1}}$, $n\ge 3$, we study the scale-invariant curvature anisotropy $A=1-f_s^2+f{f}_{ss}$ introduced by Angenent–Knopf. At the critical power $p_{\ast }=(n-1)/(n-2)$, with $\Phi =sgn(A)|A{|}^{p_{\ast }/2}$, we prove the distributional identity $$({\partial }_t-\Delta )|A{|}^{p_{\ast }}=-\frac{4}{n-1}{\left|{\Phi }_s+(n-1)\frac{f_s}{f}\Phi \right|}^2.$$ Integration against a conjugate heat kernel gives an anisotropic entropy formula. As an application, we obtain an entropy threshold for Type-I singularities on ${\mathbb{S}}^n$: 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.

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.

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 $\left|\nabla R\right|=o(f^{3/2})$ implies $R=o(f)$.

We introduce and study a Fisher information metric $g_{\tau }^F$ associated to the conjugate heat kernel of a Ricci flow $(M^n,g_t)$. This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that $g_{\tau }^F$ is monotone in scale and satisfies $g_{\tau }^F\le g_t$. We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect $g_t-g_{\tau }^F.$ This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities $0<g_{\tau }^F<g_t$ at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for $\phi$-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to $0$ to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's $\epsilon$-regularity theorem.

For a closed connected Riemannian manifold $(M,g)$, the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot )d{\mathrm{vol}}_g$. The resulting family ${\tilde{g}}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $${\tilde{g}}_t=g-2t{\mathrm{Ric}}_g+t^2\left(-\Delta {\mathrm{Ric}}_g+2{\mathrm{Ric}}_g^2-\frac{2}{3}{\mathcal{Q}}_g\right)+O_{C^0}(t^3),$$ where ${\mathcal{Q}}_g$ is quadratic in the full curvature tensor. The term $-\Delta {\mathrm{Ric}}_g$ 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 ${\tilde{g}}_t$ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is $O(t^2)$, and round spheres show that this estimate is sharp.

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 $t_0>0$ and any connected component $Z_{t_0}^{\prime }$ of the time-slice $Z_{t_0}$ of the completion, we prove that $$(\overline{{\mathcal{R}}_t^{\prime }},d_{g_t})\underset{t\nearrow t_0}{\overset{\mathrm{Gromov}\text{-}\mathrm{Hausdorff}}{\to }}(Z_{t_0}^{\prime },d_{t_0}^Z),$$ where ${\mathcal{R}}_t^{\prime }$ is the corresponding connected component of the regular part and $\overline{{\mathcal{R}}_t^{\prime }}$ 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.

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 $H_n$-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.

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.

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.

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 $\lambda$-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.

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.

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.

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.

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.

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.

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.

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 $\text{IF}$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.

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

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.