Super Ricci Flow Overview

• Super Ricci flow is a one-sided evolution inequality for time-dependent Riemannian metrics, generalizing Hamilton’s Ricci flow by allowing strict inequalities and weak formulations.
• It provides a versatile framework applicable to smooth manifolds, weighted spaces, metric-measure spaces, and discrete models, underpinning heat-flow estimates, entropy convexity, and transport contraction.
• The concept supports advanced analytic techniques such as reduced geometry, compactness methods, and singularity stratification, unifying geometric evolution across diverse settings.

Super Ricci flow is the one-sided evolution inequality

$\partial_t g_t + 2\,\Ric(g_t)\ge 0$

for a time-dependent family of Riemannian metrics, or equivalently, in backward time $\tau=T-t$,

$\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$

When the inequality is saturated to equality, one recovers Hamilton’s Ricci flow. In the literature, the notion functions both as a smooth geometric relaxation of Ricci flow and as a weak or synthetic framework for evolving metric-measure spaces, weighted manifolds, graphs, and singular spacetimes. Its principal characterizations involve transport contraction, entropy convexity, heat-flow gradient estimates, and monotonicity formulas, and in several settings it is stable under measured Gromov–Hausdorff-type limits (Lakzian, 11 Apr 2026, Sturm, 2016, Erbar et al., 2018).

1. Smooth differential-geometric formulation

On a compact $n$-manifold $M$, a smooth one-parameter family of metrics $g(t)$ is a super Ricci flow if

$\partial_t g\ge -2\,\Ric.$

In one convention used by Kunikawa–Sakurai, one writes

$h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$

so the condition becomes $h\ge -\Ric$ (Kunikawa et al., 2022). In backward-time formulations, one instead writes

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$

and the inequality becomes $\tau=T-t$0 (Kunikawa et al., 2021). These sign changes are purely conventional; they reflect time orientation rather than different geometric content.

A central refinement is Müller’s quantity. In the compact smooth setting of Kunikawa–Sakurai it is defined, for a time-dependent vector field $\tau=T-t$1, by

$\tau=T-t$2

The nonnegativity condition $\tau=T-t$3 for all $\tau=T-t$4 implies in particular the super Ricci flow inequality $\tau=T-t$5 (Kunikawa et al., 2022). In related work on almost splitting and quantitative stratification, the same structural hypothesis appears as $\tau=T-t$6 and is used to control scalar curvature, Harnack-type terms, and entropy defects (Kunikawa et al., 2023).

The smooth theory includes ancient and backward super Ricci flows. For $\tau=T-t$7, an ancient solution satisfies

$\tau=T-t$8

and one may also consider backward $\tau=T-t$9-super Ricci flow in the form

$\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$0

This setting supports Perelman-style reduced geometry, including the $\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$1-length, reduced distance $\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$2, and reduced volume $\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$3 (Kunikawa et al., 2021).

A recurrent point of interpretation is that super Ricci flow is not simply “Ricci flow with low regularity.” In the smooth category it already permits strict inequality, so it contains genuine supersolutions as well as exact Ricci flows. Lakzian’s 2026 weak-formulation paper makes this explicit: when the inequality is strict, one says $\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$4 is a super Ricci flow, while equality recovers the classical PDE (Lakzian, 11 Apr 2026).

2. Weighted and Perelman-type generalizations

A weighted version replaces the Ricci tensor by Bakry–Émery or Witten-type curvature. For a time-dependent potential $\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$5 and Witten Laplacian

$\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$6

Li–Li define a $\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$7-super Perelman Ricci flow by

$\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$8

equivalently

$\partial_\tau g(\tau)\le 2\,\Ric(g(\tau)).$9

More generally, with the $n$0-dimensional Bakry–Émery Ricci tensor

$n$1

the inequality

$n$2

is called the $n$3-super Ricci flow (Li et al., 2014, Li et al., 2017).

In this weighted setting, the super Ricci condition is tied to heat-semigroup estimates, entropy monotonicity, and differential Harnack inequalities. Li–Li prove logarithmic Sobolev inequalities, Hamilton Harnack inequalities for the heat semigroup of the Witten Laplacian, and $n$4-entropy formulas on complete or compact manifolds under $n$5-super Perelman Ricci flow and $n$6-super Ricci flow assumptions (Li et al., 2014). In the compact time-dependent case, the additional mass-conservation constraint

$n$7

ensures that the weighted measure remains fixed and allows the time-dependent $n$8-entropy formulas to extend verbatim (Li et al., 2014).

The survey of S. Li and X.-D. Li packages these facts in the language of curvature-dimension theory. If $n$9 is a $M$0-super Ricci flow and $M$1 solves the heat equation, then the associated $M$2-entropy

$M$3

obeys an entropy dissipation identity with nonnegative square and curvature terms, implying

$M$4

under the super Ricci hypothesis (Li et al., 2017).

A further analytic extension appears in work on generalized nonlinear heat-type equations along $M$5 super-Perelman Ricci flow,

$M$6

where local and global gradient estimates are proved for positive solutions of

$M$7

The resulting estimates yield Harnack-type inequalities and Liouville-type theorems (Li et al., 2024). This suggests that super Ricci flow is not merely a geometric relaxation; it is also a robust analytic background for nonlinear parabolic PDE.

3. Synthetic and weak metric-measure formulations

Sturm’s synthetic theory replaces the smooth tensor inequality by a dynamic convexity property of Boltzmann entropy on a time-dependent Wasserstein space. For a family $M$8 of Polish metric-measure spaces with

$M$9

the space is a super-Ricci flow if for almost every $g(t)$0 and every $g(t)$1 with finite entropy there exists a $g(t)$2-geodesic $g(t)$3 such that

$g(t)$4

The same theory admits an equivalent Bakry–Émery $g(t)$5-calculus formulation,

$g(t)$6

equivalent to the gradient estimate

$g(t)$7

An enforced upper-dimension version adds a positive entropy gap term and yields the notion of super-$g(t)$8-Ricci flow (Sturm, 2016).

In the smooth weighted-manifold case, this synthetic condition is equivalent to

$g(t)$9

and in the unweighted case to

$\partial_t g\ge -2\,\Ric.$0

Thus the synthetic theory recovers the classical inequality exactly (Sturm, 2016).

Lakzian’s 2026 weak-formulation paper gives two further metric-measure characterizations of smooth compact super Ricci flows. Formulation A uses heat-flow propagation of Lipschitz data and requires

$\partial_t g\ge -2\,\Ric.$1

for almost every $\partial_t g\ge -2\,\Ric.$2, where $\partial_t g\ge -2\,\Ric.$3. Formulation B, under positive scalar curvature or a “virtually psc” hypothesis, uses contraction of dynamic diffusions: $\partial_t g\ge -2\,\Ric.$4 For $\partial_t g\ge -2\,\Ric.$5, this recovers the $\partial_t g\ge -2\,\Ric.$6-Wasserstein contraction of McCann–Topping (Lakzian, 11 Apr 2026).

The same work introduces a saturation condition based on volume asymptotics of small balls and time derivatives of squared distance: $\partial_t g\ge -2\,\Ric.$7 with

$\partial_t g\ge -2\,\Ric.$8

On a compact manifold, a super Ricci flow is saturated if and only if it satisfies

$\partial_t g\ge -2\,\Ric.$9

that is, if and only if it is a genuine Ricci flow (Lakzian, 11 Apr 2026). This resolves a frequent ambiguity in weak formulations: the inequality alone encodes super Ricci flow, while saturation detects the equality case.

4. Discrete models, disjoint unions, and singular spacetimes

Super Ricci flow also has discrete and topologically singular realizations. For time-dependent finite weighted graphs $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$0, Erbar–Kopfer define a discrete super Ricci flow by any of several equivalent conditions. On nonsingular intervals, one has a forward heat operator $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$1 and dual heat operator $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$2; at singular times, collapsing blocks are handled via an explosion-integrability condition ensuring equilibration before the singular time. The discrete dynamic Bochner inequality

$h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$3

is equivalent to a two-point gradient estimate, to contraction of the discrete transport distance under dual heat flow, and to dynamic convexity of relative entropy (Erbar et al., 2018). The paper further proves consistency with continuum super Ricci flow in a discrete-to-continuum limit.

A related construction treats disjoint unions of Ricci-flowing manifolds. If $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$4 and the distance $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$5 restricts to the intrinsic Riemannian distances on each component, Lakzian–Munn define $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$6 to be a super Ricci flow if the Lipschitz constant of every heat solution is non-increasing in time. A sufficient condition is that, for $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$7 and $h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$8,

$h=\tfrac12\,\partial_t g,\qquad H=\mathrm{tr}_g h,$9

and in particular equality by the product heat equation implies the super Ricci flow property (Lakzian et al., 2012). This provides a purely metric mechanism for passing through topological disconnection.

Weak super Ricci flow through neckpinch singularities is formulated in a metric-measure spacetime setting via convex transport costs. For a nondecreasing convex cost $h\ge -\Ric$0, a weak super Ricci flow requires that for conjugate-heat solutions supported in the same maximal diffusion component, the total cost

$h\ge -\Ric$1

is nonincreasing in time. In the nondegenerate spherical neckpinch setting, this property is equivalent to single-point pinching: the spacetime is a weak super Ricci flow for a convex cost if and only if the singularity occurs on a finite number of totally geodesic hypersurfaces of the form $h\ge -\Ric$2, rather than along a positive-length interval (Lakzian et al., 2020).

These examples show that super Ricci flow is not confined to fixed smooth manifolds. Graphs with changing combinatorics, disjoint unions, and singular neckpinch continuations all fit within its functional-analytic core, provided the relevant heat-flow and transport structures remain controlled.

5. Heat kernels, reduced geometry, and regularity estimates

A major analytic development is the extension of Bamler–Zhang heat-kernel technology to super Ricci flow with nonnegative Müller quantity. Under the standing hypothesis

$h\ge -\Ric$3

Kunikawa–Sakurai recover several Ricci-flow tools: monotonicity of Perelman’s $h\ge -\Ric$4-functional, a uniform logarithmic Sobolev inequality, a uniform Sobolev inequality, $h\ge -\Ric$5-noncollapsing under scalar-curvature bounds, existence and estimates for the heat kernel $h\ge -\Ric$6, the reduced distance barrier inequality

$h\ge -\Ric$7

and the lower bound

$h\ge -\Ric$8

They also obtain the Zhang-type gradient estimate for positive heat solutions

$h\ge -\Ric$9

(Kunikawa et al., 2022).

The principal Gaussian heat-kernel theorem states that, fixing $h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$0, there exist constants $h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$1 such that whenever $h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$2 and

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$3

the heat kernel satisfies

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$4

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$5

and

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$6

The proof follows the Bamler–Zhang blueprint via distance distortion, parabolic cutoffs, Moser iteration for the conjugate heat equation, and a near-diagonal/far-off-diagonal decomposition (Kunikawa et al., 2022).

Reduced geometry on ancient super Ricci flows supplies a complementary analytic framework. Under additional Müller–Hamilton-type assumptions

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$7

one has

$h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$8

and the reduced volume $h=-\,\partial_\tau g,\qquad H=\mathrm{tr}_g h,$9 is monotone non-increasing (Kunikawa et al., 2021). This machinery is used to prove Liouville theorems for harmonic map heat flow along ancient super Ricci flow, with sharp growth conditions for non-positively curved targets and new results even in the static case for certain positively curved targets (Kunikawa et al., 2021).

A plausible implication is that the analytic identity of super Ricci flow is largely determined by which Ricci-flow estimates survive under the inequality. In the literature summarized here, the decisive hypotheses are not merely $\tau=T-t$00, but strengthened structures such as $\tau=T-t$01, weighted Bakry–Émery bounds, or entropy-convexity axioms.

6. Compactness, stratification, and singularity analysis

Bamler’s compactness theory treats sequences of pointed super Ricci flows of fixed dimension and shows subsequential convergence to synthetic “metric flow” limits. Given pointed flows $\tau=T-t$02 on $\tau=T-t$03, the associated metric-flow pairs

$\tau=T-t$04

are precompact in the $\tau=T-t$05-distance. After passing to a subsequence,

$\tau=T-t$06

where the limit is an $\tau=T-t$07-concentrated metric flow pair (Bamler, 2020).

A metric flow consists of time slices $\tau=T-t$08 together with backward transition measures $\tau=T-t$09 satisfying normalization, a reproduction formula, and a one-dimensional heat-flow gradient bound. Under local curvature bounds and noncollapsing for approximating Ricci flows, the limit decomposes into a regular set $\tau=T-t$10 and a singular set $\tau=T-t$11: $\tau=T-t$12 is open and carries a smooth Ricci-flow spacetime structure, while $\tau=T-t$13 has parabolic Hausdorff dimension at most $\tau=T-t$14 and parabolic Minkowski codimension at least $\tau=T-t$15 (Bamler, 2020).

Kunikawa–Sakurai extend Bamler’s almost rigidity theory from Ricci flow to super Ricci flow under the hypothesis $\tau=T-t$16. They define $\tau=T-t$17-selfsimilar points using the conjugate heat kernel measure and smallness of the defect

$\tau=T-t$18

prove an almost splitting theorem, and establish quantitative stratification bounds of the form

$\tau=T-t$19

They also obtain almost constancy of $\tau=T-t$20 at almost selfsimilar points, described in the paper as new even for Ricci flow (Kunikawa et al., 2023).

The same circle of ideas connects super Ricci flow to singularity models and compactness beyond the smooth manifold category. The 2026 weak-formulation program argues that the notions of WSRF, $\tau=T-t$21-WSRF, and saturation require only metric-measure data, formal Trotter–Chernoff heat propagation, local Lipschitz constants or cost contraction, and volume asymptotics of small balls, so they extend straightforwardly to singular compact metric-measure spaces with mild regularity (Lakzian, 11 Apr 2026). Sturm’s earlier synthetic theory likewise proves stability under space-time measured Gromov–Hausdorff convergence and precompactness for uniformly bounded families of super-$\tau=T-t$22-Ricci flows (Sturm, 2016).

The range of examples is correspondingly broad. Beyond ordinary Ricci flow, the Müller-based heat-kernel theory applies to List flow, Müller flow, twisted Kähler–Ricci flow, and the mean-curvature flow of space-like hypersurfaces in nonnegatively curved Lorentzian manifolds whenever the corresponding Müller quantity is known to be nonnegative (Kunikawa et al., 2022). In this sense, super Ricci flow serves both as a relaxation of Ricci flow and as a unifying analytic category for geometric evolution inequalities across smooth, weighted, synthetic, discrete, and singular settings.