on three proofs of Gromov's volume growth conjecture

Overview

Three August 2026 preprints give independent proofs of Gromov’s codimension-two volume-growth conjecture. Jian Ge combines a heat-kernel Fisher metric with Nash entropy (Ge, 2026). Bochao Kong and Xingyu Zhu use a heat-kernel Wasserstein metric, splitting rigidity, and a count of long minimizing geodesics (Kong & Zhu, 2026). Gioacchino Antonelli instead combines harmonic almost-splitting maps, a quantitative Hodge obstruction, and an inductive improvement of the effective Euclidean rank (Antonelli, 2026).

The three arguments share a geometric principle: positive scalar curvature prevents a space with nonnegative Ricci curvature from looking Euclidean in all but one direction over arbitrarily large scales. Ge and Kong—Zhu detect the missing directions through heat flow and convert the defect into volume loss by entropy decay or midpoint pair-counting. Antonelli detects the same obstruction through the Weitzenböck curvature term on differential forms and propagates it across scales by rank improvement. Kong—Zhu also prove Gromov’s separate codimension-one prediction for a uniform unit-ball volume deficit, while Antonelli proves a stronger family of bounds for positive intermediate curvature.

Preprint status and provenance

Ge’s preprint is arXiv v1 from August 13, 2026, and Kong—Zhu’s is v1 from August 14. Antonelli’s preprint first appeared on August 14 and was revised to v2 on August 19. The discussion below therefore describes recent preprints rather than a peer-reviewed consensus.

Kong and Zhu disclose that ChatGPT 5.6 Sol Ultra and Codex assisted proof exploration, organization, and drafting, that essential ideas were generated by AI, and that the authors reviewed the work and remain responsible for it (Kong & Zhu, 2026). Antonelli separately discloses substantial use of the same systems: GPT proposed the central Hodge-obstruction/rank-improvement induction; Antonelli states that he formulated and guided the problem, developed the effective argument and intermediate-curvature extension, and refined parts of the proof through discussions with collaborators (Antonelli, 2026). The manuscripts describe the three works as independent.

🏷️ Geometric Preliminaries

Let $(M^n,g)$ be a complete, connected Riemannian manifold. The common Gromov theorem concerns the noncompact case, while Antonelli’s general theorem does not require noncompactness. With the convention

$$\mathrm{Ric}(v,v)=\sum _{i=1}^n⟨R(e_i,v)v,e_i⟩,\mathrm{Scal}={\mathrm{tr}}_g\mathrm{Ric},$$

the condition $\mathrm{Ric}\ge 0$ means that the Ricci tensor is nonnegative as a quadratic form. It implies $\mathrm{Scal}\ge 0$, but the lower bound $\mathrm{Scal}\ge \kappa$ controls only the trace of Ricci. In particular, it does not give a positive lower bound for every Ricci eigenvalue. Producing two macroscopic defective directions from this trace information is the central difficulty of the codimension-two theorem.

To see the issue algebraically, diagonalize $\mathrm{Ric}$ at a point:

$$0\le {\lambda }_1\le \cdots \le {\lambda }_n,\mathrm{Scal}={\lambda }_1+\cdots +{\lambda }_n.$$

The scalar lower bound only implies ${\lambda }_n\ge \kappa /n$. It does not force ${\lambda }_{n-1}>0$: the eigenvalue pattern

$$(0,\dots ,0,\kappa )$$

is compatible with the pointwise trace inequalities. Thus the second lost direction cannot come from a pointwise statement that two Ricci eigenvalues are positive. Each proof instead couples curvature to a spatial or parabolic rigidity mechanism.

Scalar curvature first appears in the small-ball expansion

$$\mathrm{Vol}B(p,r)={\omega }_n{r}^n\left(1-\frac{\mathrm{Scal}(p)}{6(n+2)}{r}^2+O(r^4)\right)(r\downarrow 0).$$

Thus positive scalar curvature creates a second-order local volume deficit. Gromov’s question asks whether a uniform local curvature bound continues to remove two powers of radius at arbitrarily large scales. The small-ball expansion alone cannot make that passage.

☘️ Intermediate curvature

Antonelli’s theorem is naturally stated using the intermediate curvatures introduced by Brendle, Hirsch, and Johne (Brendle, Hirsch & Johne, 2024). For an $r$-plane $W\subset T_{x}M$, choose an orthonormal basis $e_1,\dots ,e_r$ and extend it to an orthonormal basis of $T_{x}M$. Define

$${\mathcal{C}}_r(W)=\sum _{a=1}^r\sum _{b=a+1}^n\mathrm{Sect}(e_a\wedge e_b).$$

Equivalently,

$${\mathcal{C}}_r(W)=\sum _{1\le a<b\le r}{K}_{ab}+\sum _{\begin{array}{c}1\le a\le r\\ r<b\le n\end{array}}{K}_{ab},K_{ab}=\mathrm{Sect}(e_a\wedge e_b).$$

The first sum records planes internal to $W$, while the second records planes with one direction in $W$ and one normal to it. This expression is independent of the chosen orthonormal basis and extension. The notation

$${\mathcal{C}}_r\ge c$$

means ${\mathcal{C}}_r(W)\ge c$ for every $r$-plane $W\subset T_{x}M$ and every point $x$. At the two endpoints relevant here,

$${\mathcal{C}}_1(\mathrm{span}\{e_1\})=\mathrm{Ric}(e_1,e_1),{\mathcal{C}}_{n-1}(e_n^{\perp })=\frac{1}{2}\mathrm{Scal}(x).$$

The case ${\mathcal{C}}_2$ is often called biRicci curvature. Thus the conditions ${\mathcal{C}}_{m+1}>0$, for $0\le m\le n-2$, interpolate between positive Ricci curvature and positive scalar curvature. The index is aligned with the desired growth exponent: positivity of ${\mathcal{C}}_{m+1}$ obstructs $m+1$ Euclidean splitting directions and leads to growth of order at most $R^m$.

Here $m$ is the predicted effective growth dimension, not the codimension. To prove growth of order $R^m$, one must rule out $m+1$ independent Euclidean directions. The number of powers lost from the ambient $R^n$ bound is $n-m$. In the scalar-curvature case $m=n-2$, so the required obstruction is precisely an $(n-1)$-fold splitting.

For $p\in M$, write

$$V_p(R)={\mathrm{Vol}}_{g}B(p,R).$$

The uniform ball-volume function is

$$V(M,R)=\underset{p\in M}{\sup }{V}_p(R).$$

For an exponent $0\le m\le n$, it is useful to remove the expected power of the radius and write

$${\mathcal{E}}_m(p,R)=\frac{\mathrm{Vol}B(p,R)}{R^m}.$$

A uniform $R^m$ growth estimate is exactly a uniform upper bound for ${\mathcal{E}}_m$. Bishop—Gromov says that ${\mathcal{E}}_n(p,R)$ is nonincreasing when $\mathrm{Ric}\ge 0$. For $m<n$, no such monotonicity is available; Antonelli’s rank-improvement argument will instead compare ${\mathcal{E}}_m$ on a sequence of carefully selected balls.

If ${\mathrm{Ric}}_g\ge 0$, Bishop—Gromov comparison gives

$$V_p(R)\le {\omega }_n{R}^n,$$

uniformly in the center $p$. More precisely, $R\mapsto V_p(R)/R^n$ is nonincreasing. This is the Euclidean growth bound.

☘️ Curvature and metric scaling

For a constant $c>0$, put $\tilde{g}=cg$. Then

$$d_{\tilde{g}}=\sqrt{c}{d}_g,d{\mathrm{Vol}}_{\tilde{g}}=c^{n/2}d{\mathrm{Vol}}_g,{\mathrm{Scal}}_{\tilde{g}}=c^{-1}{\mathrm{Scal}}_g.$$

Taking $c=\kappa$ turns ${\mathrm{Scal}}_g\ge \kappa$ into ${\mathrm{Scal}}_{\kappa g}\ge 1$. If the normalized theorem gives

$${\mathrm{Vol}}_{\kappa g}{B}_{\kappa g}(p,\sqrt{\kappa }R)\le C_n(\sqrt{\kappa }R{)}^{n-2},$$

then

$${\kappa }^{n/2}{\mathrm{Vol}}_g{B}_g(p,R)\le C_n{\kappa }^{(n-2)/2}{R}^{n-2}.$$

After division by ${\kappa }^{n/2}$, the scale-covariant factor is ${\kappa }^{-1}$. This explains why the theorem contains ${\kappa }^{-1}$ rather than ${\kappa }^{-1/2}$.

Antonelli writes the intermediate-curvature lower bound as

$${\mathcal{C}}_{m+1}\ge {\sigma }^2,$$

where $\sigma$ has units of inverse length. The corresponding scale-covariant factor is

$${\sigma }^{-(n-m)}{R}^m.$$

For the scalar-curvature specialization, $\mathrm{Scal}\ge \kappa$ means ${\mathcal{C}}_{n-1}\ge \kappa /2$, so ${\sigma }^2=\kappa /2$ and the factor ${\sigma }^{-2}=2{\kappa }^{-1}$ agrees with the normalization above after absorbing the factor $2$ into the dimensional constant.

☘️ Codimension by polynomial growth

A uniform estimate

$$V_p(R)\le C{R}^{n-k}$$

will be called codimension-$k$ volume growth: at large scales, at least $k$ of the original $n$ directions no longer contribute a full power of $R$.

Gromov’s 1986 problem predicts two sources of such dimension loss (Gromov, 1986). A definite deficit from Euclidean volume at one fixed scale should force codimension one, while a definite positive scalar-curvature lower bound should force codimension two. These exponents are sharp:

• A thin flat cylinder $S_a^1\times {\mathbb{R}}^{n-1}$ has a uniform unit-ball volume deficit and volume growth of order $R^{n-1}$.

• The product $S_a^2\times {\mathbb{R}}^{n-2}$ satisfies

  $$\mathrm{Ric}\ge 0,\mathrm{Scal}=\frac{2}{a^2},$$

  and

  $$\mathrm{Vol}B(p,R)\sim 4\pi a^2{\omega }_{n-2}{R}^{n-2}.$$

The second model explains both the exponent $n-2$ and the factor inverse to the scalar-curvature scale.

🏷️ The Common Theorem and Stronger Variants

Gromov's codimension-two volume-growth conjecture

Let $n\ge 3$. Suppose $(M^n,g)$ is complete and noncompact and

$${\mathrm{Ric}}_g\ge 0,{\mathrm{Scal}}_g\ge \kappa >0.$$

Then there is a dimensional constant $C_n$ such that

$$\mathrm{Vol}B(p,R)\le C_n{\kappa }^{-1}{R}^{n-2}$$

for every $p\in M$ and every $R>0$. Ge, Kong—Zhu, and Antonelli prove this statement by three independent methods (Ge, 2026; Kong & Zhu, 2026; Antonelli, 2026).

For $n=2$, the scalar lower bound is a positive Ricci lower bound, so Bonnet—Myers rules out the complete noncompact case. The substantive problem begins in dimension three.

Kong--Zhu's codimension-one companion theorem

Let $n\ge 2$ and $\epsilon >0$. Suppose $(M^n,g)$ is complete and noncompact, ${\mathrm{Ric}}_g\ge 0$, and

$$\underset{x\in M}{\sup }\mathrm{Vol}B(x,1)\le {\omega }_n-\epsilon .$$

Then there is $C=C(n,\epsilon )<\infty$ such that

$$\mathrm{Vol}B(p,R)\le C{R}^{n-1}$$

for every $p\in M$ and $R>0$ (Kong & Zhu, 2026).

Antonelli's intermediate-curvature theorem

Let $n\ge 2$ and $0\le m\le n-2$. There are constants $\nu (n,m),C(n,m)>0$ with the following property. Suppose $(M^n,g)$ is complete and connected and, for $\delta \ge 0$ and $\sigma >0$,

$${\mathrm{Ric}}_g\ge -{\delta }^2,{\mathcal{C}}_{m+1}\ge {\sigma }^2.$$

If $\delta R\le \nu (n,m)$, then for every center $p$,

$$\mathrm{Vol}B(p,R)\le C(n,m){\sigma }^{-(n-m)}{R}^m.$$

When $\delta =0$, the conclusion holds at every radius. Taking $m=n-2$ and ${\sigma }^2=\kappa /2$ recovers the common codimension-two theorem (Antonelli, 2026).

Proof	Quantitative detector	Curvature obstruction	Conversion to volume
Ge	Fisher trace defect and its parabolic source	A heat-averaged two-unit production estimate	Nash entropy decay followed by Li—Yau
Kong—Zhu	Eigenvalues of the Wasserstein pullback-metric deficit	Exclusion of an $(n-1)$-fold harmonic almost splitting	Long-geodesic scarcity and ordered-pair counting
Antonelli	Effective Euclidean rank of a ball and its $m$-volume ratio	Hodge obstruction to an $(m+1)$-fold harmonic splitting	Critical-scale rank improvement iterated $m+1$ times

Proof architecture

Each proof separates into a detector, a curvature obstruction, and a volume converter.

• Ge’s detector is scalar: the Fisher trace defect. Curvature supplies two units to its parabolic source, and two heat-potential integrations turn those units into entropy decay.
• Kong—Zhu’s detector is directional: the eigenvalues of $D_t^W$. Curvature forces two positive deficit eigenvalues, and phase-space counting converts two angular restrictions into two missing powers of radius.
• Antonelli’s detector is geometric: the largest Euclidean rank visible on a ball. Intermediate curvature bounds the scale of an $(m+1)$-split ball, and rank improvement transports an $m$-volume ratio to such a terminal ball.

The recurring scale dictionary is

$$\text{spatial radius }R⟷\text{heat time }t=R^2⟷\text{curvature scale }{\kappa }^{-1/2}.$$

🏷️ Heat-Kernel, Splitting, and Hodge Preliminaries

☘️ Heat kernel and semigroup

The sign convention is $\Delta =\mathrm{div}\nabla$, so $-\Delta$ is nonnegative on $L^2(M)$. The minimal heat kernel is the positive solution of

$$({\partial }_t-{\Delta }_x)H(x,y,t)=0$$

with initial data ${\delta }_y$. Under $\mathrm{Ric}\ge 0$, it is symmetric and conservative:

$$H(x,y,t)=H(y,x,t),{\int }_{M}H(x,y,t)d{\mathrm{Vol}}_g(y)=1.$$

It satisfies the semigroup identity

$$H(x,y,t+s)={\int }_{M}H(x,z,t)H(z,y,s)d{\mathrm{Vol}}_g(z),$$

and

$$P_{t}f(x)={\int }_{M}H(x,y,t)f(y)d{\mathrm{Vol}}_g(y).$$

It is convenient to regard

$$d{\mu }_{t,x}(y)=H(x,y,t)d{\mathrm{Vol}}_g(y)$$

as a probability measure. Then $P_{t}f(x)$ is the expectation of $f$ under ${\mu }_{t,x}$.

In Euclidean space,

$$H_{{\mathbb{R}}^n}(x,y,t)=(4\pi t{)}^{-n/2}\exp \left(-\frac{|x-y{|}^2}{4t}\right).$$

The diffusion scale is therefore $\sqrt{t}$. Two estimates under $\mathrm{Ric}\ge 0$ recur in the proofs. For $d_a(z)=d(a,z)$, Laplacian comparison implies

$$P_t(d_a^2)(z)\le d(a,z{)}^2+2nt.$$

Taking $a=z=x$ gives

$${\int }_{M}d(x,y{)}^{2}d{\mu }_{t,x}(y)\le 2nt.$$

Thus a heat distribution at time $t$ sees geometry mainly within distance $O(\sqrt{t})$. A scalar lower bound $\mathrm{Scal}\ge \kappa$ has curvature length ${\kappa }^{-1/2}$, so the corresponding heat time is $t\asymp {\kappa }^{-1}$. This is why Kong—Zhu test their transport defect at time ${\kappa }^{-1}$, while Ge later takes $t=R^2$ when converting entropy at diffusion scale $\sqrt{t}$ into a bound for $B(x,R)$.

The Li—Yau theory gives a Gaussian upper bound and an integrated Fisher-information estimate (Li & Yau, 1986):

$$H(x,y,t)\le \frac{C_n}{\mathrm{Vol}B(x,\sqrt{t})}\exp \left(-\frac{d(x,y{)}^2}{5t}\right),$$

and

$${\int }_{M}H(x,y,t)|{\nabla }_y\log H(x,y,t){|}^{2}d{\mathrm{Vol}}_g(y)\le \frac{n}{2t}.$$

The numerical constant $5$ in the Gaussian exponent is inessential; what matters is a dimensional Gaussian tail uniform in the base point.

☘️ Bochner identity and heat-flow contraction

For a smooth function $u$, the Bochner identity reads

$$\frac{1}{2}\Delta |\nabla u{|}^2=|\mathrm{Hess}u{|}^2+⟨\nabla u,\nabla \Delta u⟩+\mathrm{Ric}(\nabla u,\nabla u).$$

When $\mathrm{Ric}\ge 0$, integration of this identity along the heat flow gives the Bakry—Émery gradient contraction and reverse local Poincaré inequality (Bakry, Gentil & Ledoux, 2014):

$$|\nabla P_{t}f{|}^2\le P_t|\nabla f{|}^2,$$ $$P_t(f^2)-(P_{t}f{)}^2\ge 2t|\nabla P_{t}f{|}^2.$$

The gradient estimate follows from a useful interpolation that also explains its rigidity. For fixed $t$, set

$$F(s)=P_s\left({\left|\nabla P_{t-s}f\right|}^2\right),0\le s\le t.$$

Differentiating with the Bochner identity gives

$$F^{\prime }(s)=2P_s\left({\left|\mathrm{Hess}{P}_{t-s}f\right|}^2+\mathrm{Ric}\left(\nabla P_{t-s}f,\nabla P_{t-s}f\right)\right)\ge 0.$$

Since $F(0)=|\nabla P_{t}f{|}^2$ and $F(t)=P_t|\nabla f{|}^2$, the contraction follows. Integrating the same nonnegative production term, and polarizing when several functions are present, measures precisely how far the heat flow is from carrying affine Euclidean coordinates.

The inequalities play two roles. They show that both heat-induced metrics below are bounded above by $g$, and their equality cases are rigid. Indeed, the polarized interpolation formula contains the nonnegative integrand

$$|\mathrm{Hess}{P}_{t-s}f{|}^2+\mathrm{Ric}(\nabla P_{t-s}f,\nabla P_{t-s}f).$$

If the Bakry—Émery deficit vanishes, this integrand vanishes and the limiting functions behave like affine coordinates on Euclidean factors. This is the analytic origin of the splitting step (Ambrosio, Bruè & Semola, 2019).

☘️ Harmonic almost-splitting maps

The metric precursor is quantitative Gromov—Hausdorff splitting. A ball $B_r(x)$ is $(k,\epsilon )$-split if, for some pointed proper length space $(Z,z)$, the much larger ball $B_{{\epsilon }^{-1}r}(x)$ admits a pointed $\epsilon r$-Gromov—Hausdorff approximation to the corresponding ball in

$${\mathbb{R}}^k\times Z.$$

Concretely, an $\epsilon r$-approximation distorts every pairwise distance by at most $\epsilon r$ and has $\epsilon r$-dense image; in the pointed version it also respects the chosen basepoints up to that error. Thus $(k,\epsilon )$-splitting means that $k$ independent Euclidean directions are visible, with relative error $\epsilon$, on a ball much larger than the scale $r$ under study.

Using the larger comparison ball prevents boundary effects at scale $r$. Cheeger—Colding harmonic replacement turns sufficiently accurate metric splitting, under an almost nonnegative Ricci lower bound at scale $r$, into harmonic coordinate functions with controlled gradients, nearly orthonormal Gram matrix, and small Hessian energy (Cheeger & Colding, 1996; Cheeger & Colding, 1997).

On a ball $B(x,R)$, a collection of harmonic functions $h^1,\dots ,h^k$ is an $\epsilon$-almost splitting map if its gradients are uniformly bounded, almost orthonormal on average, and have small Hessian energy. A representative normalization is

$$\underset{B(x,R)}{\sup }|\nabla h^a|\le 1+\epsilon ,$$ $$\frac{1}{\mathrm{Vol}B(x,R)}{\int }_{B(x,R)}{\left|⟨\nabla h^a,\nabla h^b⟩-{\delta }^{ab}\right|}^{2}d\mathrm{Vol}\le \epsilon ,$$

and

$$\frac{R^2}{\mathrm{Vol}B(x,R)}{\int }_{B(x,R)}|\mathrm{Hess}{h}^a{|}^{2}d\mathrm{Vol}\le \epsilon .$$

The factor $R^2$ makes the Hessian condition invariant under rescaling the ball to unit radius. Such a map is an analytic approximation to projection onto an ${\mathbb{R}}^k$ factor. Its infinitesimal $k$-dimensional Jacobian is

$${\left|d{h}^1\wedge \cdots \wedge d{h}^k\right|}^2=\det {\left(⟨\nabla h^a,\nabla h^b⟩\right)}_{a,b=1}^k.$$

Hence an almost-identity Gram matrix says that the coordinate wedge has norm close to one and is nondegenerate on most of the ball. For $k=n$, standard almost-splitting and volume-convergence theorems make the ball close to a Euclidean ball both metrically and in volume. For $k=n-1$, the map supplies the $(n-1)$-form used in the codimension-two obstruction.

☘️ Hodge Laplacian and the curvature operator

On $p$-forms, the Weitzenböck formula is

$${\Delta }_H=d\delta +\delta d={\nabla }^{\ast }\nabla +q_p(\mathrm{Rm}).$$

The bundle endomorphism $q_p(\mathrm{Rm})$ converts sectional curvature into an energy term. If ${\beta }^1,\dots ,{\beta }^n$ is an orthonormal coframe, $I\subset \{1,\dots ,n\}$ has cardinality $p$, and ${\beta }^I={\bigwedge }_{a\in I}{\beta }^a$, then (Labbi, 2015)

$$⟨q_p(\mathrm{Rm}){\beta }^I,{\beta }^I⟩=\sum _{a\in I}\sum _{b\notin I}\mathrm{Sect}(e_a\wedge e_b).$$

For $p=m+1$, put

$$\Theta ={\beta }^1\wedge \cdots \wedge {\beta }^{m+1}.$$

Since $q_1(\mathrm{Rm})=\mathrm{Ric}$, separating the sectional curvatures tangent to the $(m+1)$-plane from the mixed curvatures gives the exact identity

$$⟨q_{m+1}(\mathrm{Rm})\Theta ,\Theta ⟩+\sum _{a=1}^{m+1}⟨q_1(\mathrm{Rm}){\beta }^a,{\beta }^a⟩=2{\mathcal{C}}_{m+1}.$$

Indeed, the $q_{m+1}$-term is the sum of the mixed sectional curvatures. The sum of the $m+1$ Ricci terms counts each sectional curvature internal to the plane twice and each mixed curvature once. Adding them therefore gives twice the internal-plus-mixed sum defining ${\mathcal{C}}_{m+1}$.

This identity is the algebraic heart of Antonelli’s proof. It is also the general intermediate-curvature analogue of the $(n-1)$-form computation in the Kong—Zhu scalar-curvature obstruction.

☘️ Liouville measure and polar Jacobians

Let $SM$ be the unit tangent bundle, $\pi :SM\to M$ the footpoint map, and ${\phi }_s:SM\to SM$ the geodesic flow. If $d{\omega }_x$ is surface measure on $S_{x}M$, then

$$d\lambda (v)=d{\mathrm{Vol}}_g(x)d{\omega }_x(v)$$

is Liouville measure, and ${\phi }_s$ preserves $\lambda$. This invariance allows estimates along one minimizing geodesic to be averaged over all of phase space without a Jacobian loss.

For almost every ordered pair $(p,q)$, the minimizing geodesic is unique because the diagonal and cut relation have product measure zero. Endpoint polar coordinates contain one radial Jacobian $J(v,\ell )$. Under $\mathrm{Ric}\ge 0$, radial Bishop comparison gives

$$0\le J(v,\ell )\le {\ell }^{n-1}$$

before the cut time, and the Jacobian is set to zero afterward. Recentring an endpoint parametrization at the geodesic midpoint uses Liouville invariance; it does not introduce a second polar Jacobian. This point is essential in the ordered-pair estimate.

🏷️ Two Heat-Induced Metrics

Recall the heat distribution $d{\mu }_{t,x}(y)=H(x,y,t)d{\mathrm{Vol}}_g(y)$. The two heat-kernel manuscripts study the map

$$x⟼{\mu }_{t,x},$$

but they equip the target space of probability measures with different geometries.

☘️ The Wasserstein pullback

For probability measures with finite second moment, the quadratic Wasserstein distance is

$$W_2(\mu ,\nu {)}^2=\underset{\pi \in \Pi (\mu ,\nu )}{\inf }{\int }_{M\times M}d(x,y{)}^{2}d\pi (x,y),$$

where $\Pi (\mu ,\nu )$ is the set of couplings. An absolutely continuous curve of measures has a velocity field whose $L^2$ norm is its infinitesimal $W_2$ speed. Pulling this norm back along $x\mapsto {\mu }_{t,x}$ produces a quadratic form on $T_{x}M$ (Gigli & Mantegazza, 2014).

Kong and Zhu use the following variational form of that pullback:

$$G_{t,x}^W(v,v)=\underset{\psi \in C_c^{\infty }(M)}{\sup }\left\{2⟨\nabla P_t\psi (x),v⟩-P_t|\nabla \psi {|}^2(x)\right\}.$$

The Bakry—Émery gradient contraction

$$|\nabla P_t\psi {|}^2\le P_t|\nabla \psi {|}^2$$

implies

$$0\le G_t^W\le g.$$

The nonnegative tensor

$$D_t^W:=g-G_t^W$$

measures how much the heat-distribution map contracts each tangent direction in $W_2$.

☘️ The Fisher pullback

For a positive probability density $\rho$ and a tangent perturbation $\dot{\rho }$ with integral zero, the Fisher metric is

$$g_F(\dot{\rho },\dot{\rho })={\int }_M{\left(\frac{\dot{\rho }}{\rho }\right)}^2\rho d\mathrm{Vol}.$$

Along the heat-kernel map, differentiating the pole $x$ in a direction $v$ gives the score $d_x\log H(v)$. Ge writes $H=e^{-h}$ and uses the normalized Fisher pullback

$$G_{t,x}^F=2t{\int }_M{d}_{x}h\otimes d_{x}hd{\mu }_{t,x}.$$

Reverse local Poincaré and logarithmic Sobolev inequalities again give

$$0\le G_t^F\le g.$$

Its scalar dimension defect is

$$E_x(t)=\frac{t}{2}\left(n-{\mathrm{tr}}_g{G}_{t,x}^F\right).$$

On ${\mathbb{R}}^n$, both heat metrics equal $g$. On $S_a^2\times {\mathbb{R}}^{n-2}$, the large-time Fisher metric retains the $n-2$ Euclidean directions and loses the two spherical directions, so

$${\mathrm{tr}}_g{G}_t^F⟶n-2.$$

The Euclidean normalization can be checked directly. Since

$$d_{x}h=\frac{x-y}{2t}$$

and $y-x$ is Gaussian with covariance $2tI$, one has $G_t^F=g$. The measures ${\mu }_{t,x}$ are translations of one fixed Gaussian, so

$$W_2({\mu }_{t,x},{\mu }_{t,x^{\prime }})=|x-x^{\prime }|,$$

which gives $G_t^W=g$ as well.

The scaling law makes the relevant time choices transparent. If $g_r=r^{-2}g$, so a $g$-ball of radius $r$ becomes a $g_r$-ball of radius one, then for either pullback metric

$$G_s^{\ast ,g_r}=r^{-2}{G}_{r^{2}s}^{\ast ,g},\ast =F,W.$$

Consequently,

$$D_s^{W,g_r}=r^{-2}{D}_{r^{2}s}^{W,g},$$

and the eigenvalues of $D^W$ relative to the ambient metric, as well as the trace deficit of either pullback, are unchanged by this simultaneous space—time rescaling.

Feature	Fisher route (Ge)	Wasserstein route (Kong—Zhu)
Target geometry on probability measures	Fisher information	Quadratic optimal transport
Pullback tensor	$G_t^F$	$G_t^W$
Recorded defect	Scalar trace loss, then its parabolic production	Directional eigenvalues of $D_t^W=g-G_t^W$
Final converter	Nash entropy	Long-geodesic and pair counting
Euclidean model	$G_t^F=g$	$G_t^W=g$

Distinct metrics

The Wasserstein pullback $G_t^W$ and Fisher pullback $G_t^F$ should not be identified. Their shared role is structural: each is bounded above by $g$, agrees with $g$ on Euclidean space, and records directions lost under heat flow. Their definitions, rigidity statements, and evolution identities are different.

🏷️ Proof I: Fisher Defect and Nash Entropy

Ge’s proof turns the loss of two heat directions into a logarithmic entropy deficit. The central quantity is not merely $n-\mathrm{tr}{G}_t^F$, but its parabolic production.

By the metric-scaling calculation above, it is enough to prove the result under the normalization

$$\mathrm{Scal}\ge 1.$$

Define the centered Hessian variance

$${\mathcal{V}}_x(t)={\int }_{M}H(x,y,t){\left|{\mathrm{Hess}}_{x}h(x,y,t)-\frac{G_{t,x}^F}{2t}\right|}^{2}d{\mathrm{Vol}}_g(y),$$

and the nonnegative defect source

$$Q_x(t)=4t^2{\mathcal{V}}_x(t)+|g-G_t^F{|}^2+2t⟨\mathrm{Ric},G_t^F⟩.$$

The three summands have distinct meanings. The variance term measures how far the heat-kernel score is from having a spatially constant Hessian; it vanishes for a Euclidean Gaussian. The tensor term measures the failure of the Fisher covariance to equal the ambient metric. The Ricci term charges curvature in the directions that the Fisher metric still sees. Under $\mathrm{Ric}\ge 0$, all three terms are nonnegative.

Differentiation in the pole variable $x$, together with a centered Bochner identity, yields

$$({\partial }_t-{\Delta }_x)E=\frac{1}{2}Q.$$

The differentiated variable

Although $H(x,y,t)=H(y,x,t)$, this evolution treats $x$ as the moving heat-kernel pole and integrates only in the output variable $y$. Thus ${\partial }_t-{\Delta }_x$ acts on $E_x(t)$ and later on $S_x(t)$; one must not discard $x$-divergence terms by pretending that the $y$-integration is an integration by parts in $x$.

Derivation: Centered Bochner production

Fix the output point $y$, set $u(x,t)=H(x,y,t)$, and write $h=-\log u$. The logarithmic heat equation and Bochner identity give

$$\begin{array}{rl}& ({\partial }_t-{\Delta }_x)\left[t^{2}u\left(\frac{n}{2t}-|{\nabla }_{x}h{|}^2\right)\right]\\ & =2t^{2}u\left({\left|{\mathrm{Hess}}_{x}h-\frac{g}{2t}\right|}^2+\mathrm{Ric}({\nabla }_{x}h,{\nabla }_{x}h)\right)+2t{\mathrm{div}}_x(u{\nabla }_{x}h).\end{array}$$

Integrating in $y$ is not a spatial integration by parts in $x$. Instead, stochastic completeness gives

$${\int }_{M}H(x,y,t)d{\mathrm{Vol}}_g(y)=1,$$

and differentiating this identity once and twice in the pole $x$ yields

$${\int }_{M}H{d}_{x}hd\mathrm{Vol}=0,$$ $${\int }_{M}H{\mathrm{Hess}}_{x}hd\mathrm{Vol}=\frac{G_{t,x}^F}{2t}.$$

The first equality makes the divergence term vanish. The second allows the Hessian square to be centered by the Hilbert-space variance identity

$$\int |A-\alpha {|}^{2}d\mu =\int |A-\overline{A}{|}^{2}d\mu +|\overline{A}-\alpha {|}^2,$$

with $A={\mathrm{Hess}}_{x}h$, $\overline{A}=G_t^F/(2t)$, and $\alpha =g/(2t)$. The Ricci term becomes

$$4t^2{\int }_M\mathrm{Ric}({\nabla }_{x}h,{\nabla }_{x}h)d{\mu }_{t,x}=2t⟨\mathrm{Ric},G_t^F⟩.$$

The integrated left side is exactly $E_x(t)$, and these three centered terms are exactly $Q_x(t)/2$ after accounting for the factor in the evolution equation.

The scalar-curvature assumption first gives one unit of production by a pointwise algebraic inequality. If $A\ge 0$, $0\le G\le I$, $\mathrm{tr}A\ge 1$, and $t\ge 1$, then

$$|I-G{|}^2+2t⟨A,G⟩\ge 1.$$

Indeed, if ${\lambda }_1$ is the smallest eigenvalue of $G$, then

$$⟨A,G⟩\ge {\lambda }_1\mathrm{tr}A\ge {\lambda }_1,|I-G{|}^2\ge (1-{\lambda }_1{)}^2.$$

Therefore the left side is at least $(1-{\lambda }_1{)}^2+2t{\lambda }_1\ge 1$. Taking $A={\mathrm{Ric}}^{♯}$ and $G=(G_t^F{)}^{♯}$ shows $Q\ge 1$. Obtaining the second unit is the geometric core.

Proof: The heat-averaged second unit

Fix $0<\epsilon \le 1/2$ and consider the bad region

$${\Omega }_{\epsilon }(t)=\{x:Q_x(t)<2-\epsilon \}.$$

For $t\ge 6/\epsilon$, the smallness of $Q$ forces the smallest eigenvalue of $G_t^F$ below $\epsilon /6$ and the second eigenvalue above $\epsilon /3$. Hence there is a distinguished lowest eigenline with a definite spectral gap. Positivity of Ricci and $\mathrm{Scal}\ge 1$ further imply that this eigenline carries Ricci curvature greater than $1/2$.

An orthogonally invariant matrix cutoff localizes to this eigenline without requiring it to be globally orientable. A weighted Weitzenböck estimate for the associated rank-one projection $\Pi$ gives

$$\int w{\varphi }^2⟨\mathrm{Ric},\Pi ⟩\le C_n\int w\left({\varphi }^2|\nabla \Pi {|}^2+|\nabla \varphi {|}^2+{\varphi }^2|\nabla \log w{|}^2\right),$$

where $w(x)=H(z,x,s)$. The covariance derivative estimates control the first two terms by $Q/({\epsilon }^{2}t)$, while the integrated Li—Yau estimate controls the last term by $C_n/s$. Consequently,

$$P_s{1}_{{\Omega }_{\epsilon }(t)}(z)\le C_n{\epsilon }^{-2}\left(\frac{P_s{Q}_t(z)}{t}+\frac{1}{s}\right).$$

Off the bad region, $Q\ge 2-\epsilon$. Combining these two facts and optimizing at

$$\epsilon \asymp (t^{-1}+s^{-1}{)}^{1/3}$$

produces the heat-averaged estimate

$$P_s{Q}_t(z)\ge 2-C_n(t^{-1}+s^{-1}{)}^{1/3}.$$

Thus the second unit need not hold pointwise: the argument proves that the region where it fails has asymptotically negligible heat mass.

The normalized Nash entropy is (Ni, 2004; Colding, 2012)

$$S_x(t)=-{\int }_{M}H(x,y,t)\log H(x,y,t)d{\mathrm{Vol}}_g(y)-\frac{n}{2}\log (4\pi t)-\frac{n}{2}.$$

It vanishes on Euclidean space and satisfies the second pole-variable identity

$$({\partial }_t-{\Delta }_x)(-S)=\frac{E}{t^2}.$$

The signs needed below are canonical. Since $0\le G_t^F\le g$, one has $E\ge 0$. The sharp heat-kernel entropy inequality under $\mathrm{Ric}\ge 0$ gives $S\le 0$, with equality in the Euclidean model, so $-S\ge 0$.

Applying the minimal heat-potential comparison first to $E$ and then to $-S$ propagates the two source units through the chain

$$Q⟶E⟶-S.$$

Here is the origin of the integration kernel. If a nonnegative function $u$ satisfies

$$({\partial }_t-\Delta )u\ge f,$$

then Duhamel comparison gives, for $b>a$,

$$u(b)\ge P_{b-a}u(a)+{\int }_a^b{P}_{b-s}f(s)ds.$$

Starting both equations at time $1$, discarding the nonnegative initial terms, and using the semigroup identity gives

$$\begin{array}{rl}-S_x(T)& \ge {\int }_1^T\frac{P_{T-\tau }{E}_{\tau }(x)}{{\tau }^2}d\tau \\ & \ge \frac{1}{2}{\int }_1^T{\int }_1^{\tau }\frac{P_{T-s}{Q}_s(x)}{{\tau }^2}dsd\tau .\end{array}$$

Tonelli’s theorem reverses the order of integration, and

$${\int }_s^T\frac{d\tau }{{\tau }^2}=\frac{1}{s}-\frac{1}{T}.$$

Therefore

$$-S_x(T)\ge \frac{1}{2}{\int }_1^T\left(\frac{1}{\tau }-\frac{1}{T}\right)P_{T-\tau }{Q}_{\tau }(x)d\tau .$$

The main term is explicit:

$$\frac{1}{2}{\int }_1^T\left(\frac{1}{\tau }-\frac{1}{T}\right)2d\tau =\log T-1+\frac{1}{T}.$$

The error produced by the two-unit estimate is bounded independently of $T$ because

$${\int }_1^T\left(\frac{1}{\tau }-\frac{1}{T}\right){\left({\tau }^{-1}+(T-\tau {)}^{-1}\right)}^{1/3}d\tau \le C.$$

The ${\tau }^{-1/3}$ contribution is bounded by ${\int }_1^T{\tau }^{-4/3}d\tau$. For the $(T-\tau {)}^{-1/3}$ contribution, splitting at $T/2$ gives a bound by $C{T}^{-1/3}\log T+C$. Both bounds are uniform in $T$. Substitution of the two-unit estimate therefore gives

$$S_x(T)\le -\log T+C_n,T\ge 2.$$

Finally, the Li—Yau Gaussian upper bound implies

$$-\log H(x,y,t)\ge \log \mathrm{Vol}B(x,\sqrt{t})-C_n+\frac{d(x,y{)}^2}{5t}.$$

Multiplying by $H$ and integrating in $y$, stochastic completeness gives

$$-{\int }_{M}H\log Hd\mathrm{Vol}\ge \log \mathrm{Vol}B(x,\sqrt{t})-C_n,$$

after discarding the nonnegative distance moment. Subtracting the Euclidean normalization in $S_x(t)$ gives

$$\log \frac{\mathrm{Vol}B(x,\sqrt{t})}{t^{n/2}}\le S_x(t)+C_n.$$

Therefore

$$\mathrm{Vol}B(x,\sqrt{t})\le C_n{t}^{(n-2)/2},t\ge 2.$$

For $R\ge \sqrt{2}$, take $t=R^2$ to obtain

$$\mathrm{Vol}B(x,R)\le C_n{R}^{n-2}.$$

For $0<R\le \sqrt{2}$, Bishop—Gromov gives

$$\mathrm{Vol}B(x,R)\le {\omega }_n{R}^n\le 2{\omega }_n{R}^{n-2}.$$

This completes the normalized proof at every radius; scaling back restores the factor ${\kappa }^{-1}$. Analytically, each persistent missing heat direction contributes approximately $-\frac{1}{2}\log t$ to the normalized entropy; two directions contribute the full $-\log t$ needed to remove two powers of $R$.

🏷️ Proof II: Wasserstein Defect and Geodesic Counting

Kong and Zhu use one transport engine for two conclusions. A unit-ball volume deficit produces one defective heat direction and proves the codimension-one companion theorem; positive scalar curvature produces two defective directions and proves the second of the three codimension-two arguments.

Kong and Zhu do not evolve a scalar entropy. They use $D_t^W$ to show that long minimizing geodesics cannot occupy too many directions, and then count pairs of points joined by those geodesics.

If

$$\gamma :[-3r/2,3r/2]⟶M$$

is a unit-speed minimizing geodesic, Wasserstein transport of the endpoint heat distributions gives the centered segment budget

$${\int }_{-r}^r{D}_t^W(\dot{\gamma },\dot{\gamma })ds\le 2\left(\sqrt{r^2+8nt}-r\right)\le \frac{8nt}{r}.$$

Here $D_t^W(\dot{\gamma },\dot{\gamma })$ is a dimensionless directional contraction, so its integral has units of length, matching $t/r$. A positive eigenvalue of $D_t^W$ therefore assigns a definite cost to geodesics moving in that eigendirection. The buffer from $[-3r/2,3r/2]$ to $[-r,r]$ keeps the two comparison endpoints a distance $r/2$ beyond the segment on which the defect is integrated.

Derivation: The centered segment budget

Put

$$p=\gamma (-3r/2),x=\gamma (-r),y=\gamma (r),q=\gamma (3r/2).$$

Since $d_p+d_q\ge d(p,q)=3r$, the squared-distance heat estimate and Cauchy—Schwarz give

$$P_t{d}_p(x),P_t{d}_q(y)\le \sqrt{r^2/4+2nt},$$

while $P_t{d}_p(y)\ge 3r-P_t{d}_q(y)$. Kantorovich duality for $W_1$, followed by $W_1\le W_2$, therefore yields

$$W_2({\mu }_{t,x},{\mu }_{t,y})\ge 3r-2\sqrt{r^2/4+2nt}.$$

In the other direction, the $W_2$ length of the heat-distribution curve along $\gamma {|}_{[-r,r]}$ satisfies

$$W_2({\mu }_{t,x},{\mu }_{t,y})\le {\int }_{-r}^r\sqrt{G_t^W(\dot{\gamma },\dot{\gamma })}ds.$$

Since $G_t^W(\dot{\gamma },\dot{\gamma })=1-D_t^W(\dot{\gamma },\dot{\gamma })$ and $\sqrt{1-a}\le 1-a/2$ for $0\le a\le 1$,

$$W_2({\mu }_{t,x},{\mu }_{t,y})\le 2r-\frac{1}{2}{\int }_{-r}^r{D}_t^W(\dot{\gamma },\dot{\gamma })ds.$$

Comparing the two bounds gives the first displayed estimate; rationalizing the square root gives the simpler upper bound $8nt/r$.

The lower bound comes from Kantorovich duality and heat-flow control of squared distance; the upper bound comes from measuring the same curve using $G_t^W=g-D_t^W$. The important feature is the decay $r^{-1}$: along a very long minimizing segment, the velocity has little average component in directions where $D_t^W$ is positive.

Suppose $D_t^W(x)$ has at least $k\in \{1,2\}$ eigenvalues bounded below by $\rho >0$ at every point. For a measurable set $F\subset S_{x}M$, an angular coercivity estimate gives

$${\int }_F{D}_t^W(v,v)d{\omega }_x(v)\ge c(n,k)\rho {\omega }_x(F{)}^{1+2/k}.$$

The exponent comes from the volume of a thin equatorial neighborhood perpendicular to a $k$-plane: a set of directions can avoid $k$ defective directions only by concentrating in a band whose measure is of order $a^k$.

For $k=1$, such a band has measure of order $a$ while leaving the band costs order $a^2$, which produces the cubic power ${\omega }_x(F{)}^3$. For $k=2$, the band has measure of order $a^2$, so the same quadratic cost produces ${\omega }_x(F{)}^2$. These are exactly the exponents $1+2/k$ that later become one or two missing powers of radius.

Derivation: Angular coercivity

Let $\Pi \subset T_{x}M$ be the span of $k$ eigenvectors whose eigenvalues are at least $\rho$. Then

$$D_t^W(v,v)\ge \rho |{\mathrm{proj}}_{\Pi }v{|}^2.$$

Spherical coordinates around the orthogonal sphere $S({\Pi }^{\perp })$ give a dimensional constant $b(n,k)$ such that

$${\omega }_x\{v:|{\mathrm{proj}}_{\Pi }v|\le a\}\le b(n,k)a^k(0<a\le 1).$$

Given $F\subset S_{x}M$, choose

$$a={\left(\frac{{\omega }_x(F)}{2b(n,k)}\right)}^{1/k}.$$

At least half the measure of $F$ lies outside the band, where $D_t^W(v,v)\ge \rho a^2$. Therefore

$${\int }_F{D}_t^W(v,v)d{\omega }_x(v)\ge \frac{\rho }{2(2b(n,k){)}^{2/k}}{\omega }_x(F{)}^{1+2/k}.$$

For $v\in SM$, set ${\gamma }_v(s)=\pi ({\phi }_{s}v)$ and define

$${\mathcal{G}}_L=\left\{v:{\gamma }_v{|}_{[-L/2,L/2]}\text{ is minimizing}\right\}.$$

For a Borel set $\Omega \subset M$, write $S\Omega ={\pi }^{-1}(\Omega )$ and

$$Q_{\Omega }(L)=\lambda (S\Omega \cap {\mathcal{G}}_L).$$

Also write ${\Omega }_a=\{z\in M:d(z,\Omega )<a\}$ for the open $a$-neighborhood. Thus $Q_{\Omega }(L)$ measures the Liouville mass of unit directions based in $\Omega$ which remain minimizing for length $L$ about their midpoint.

Liouville invariance lets one average the segment budget over these long minimizing directions. Hölder’s inequality in the base variable converts the angular exponent $1+2/k$ into the power $L^{-k}$, giving

$$Q_{\Omega }(L)\le K_{n,k}{\left(\frac{t}{\rho }\right)}^{k/2}\frac{\mathrm{Vol}({\Omega }_{L/3})}{L^k}.$$

Derivation: Liouville averaging

Write $L=3r$, let $E=S\Omega \cap {\mathcal{G}}_{3r}$, and set $Q=\lambda (E)$. Integration of the centered segment budget over $E$ gives

$${\int }_{-r}^r{\int }_E{D}_t^W({\phi }_{s}v,{\phi }_{s}v)d\lambda (v)ds\le \frac{8nt}{r}Q.$$

For fixed $s$, Liouville invariance gives $\lambda ({\phi }_{s}E)=Q$, and the footpoints of ${\phi }_{s}E$ lie in ${\Omega }_r$. Define the angular fiber mass

$$q_s(z)={\int }_{S_{z}M}{1}_{{\phi }_{s}E}(v)d{\omega }_z(v).$$

Then ${\int }_{{\Omega }_r}{q}_{s}d\mathrm{Vol}=Q$. Angular coercivity followed by Hölder gives

$${\int }_{{\phi }_{s}E}{D}_t^W(v,v)d\lambda (v)\ge c(n,k)\rho \frac{Q^{1+2/k}}{\mathrm{Vol}({\Omega }_r{)}^{2/k}}.$$

Integrating this lower bound over $s\in (-r,r)$, comparing it with the segment budget, canceling one factor of $Q$, and raising to the power $k/2$ gives the claimed $r^{-k}$ estimate.

Thus $k$ positive heat-deficit eigenvalues remove $k$ powers from the abundance of long minimizing segments.

The final bridge to ball volume uses ordered pairs. Almost every pair $(p,q)$ has a unique minimizing geodesic and hence a midpoint. Recenter endpoint polar coordinates at that midpoint. Bishop comparison bounds the polar Jacobian by ${\ell }^{n-1}$, while the preceding estimate contributes ${\ell }^{-k}$. Consequently,

$${\mathrm{Vol}}_{M\times M}{\mathcal{E}}_{\Omega }(R)\le K_{n,k}{\left(\frac{t}{\rho }\right)}^{k/2}{\int }_0^R{\ell }^{n-1-k}\mathrm{Vol}({\Omega }_{\ell /3})d\ell ,$$

where ${\mathcal{E}}_{\Omega }(R)$ consists of ordered pairs at distance below $R$ whose midpoint lies in $\Omega$.

Derivation: Midpoint polar coordinates

Endpoint polar coordinates state that, for nonnegative measurable $f$,

$${\int }_{M\times M}f(p,q)dpdq={\int }_{SM}{\int }_0^{\infty }f(\pi v,\pi {\phi }_{\ell }v)\overline{J}(v,\ell )d\ell d\lambda (v),$$

where $\overline{J}$ is the radial Jacobian before the cut time and zero afterward. At fixed $\ell$, change variables to the midpoint direction $w={\phi }_{\ell /2}v$. Liouville invariance preserves $d\lambda$, and the midpoint condition becomes $w\in S\Omega$. Radial Bishop comparison gives

$$\overline{J}({\phi }_{-\ell /2}w,\ell )\le {\ell }^{n-1}{1}_{{\mathcal{G}}_{\ell }}(w).$$

Hence

$${\mathrm{Vol}}_{M\times M}{\mathcal{E}}_{\Omega }(R)\le {\int }_0^R{\ell }^{n-1}{Q}_{\Omega }(\ell )d\ell .$$

Substitution of the Liouville tail bound produces the factor ${\ell }^{n-1-k}$ in the preceding display. The assumption $n>k$ makes this integral finite at $\ell =0$.

Take $\Omega =B(o,2R)$ and use the pair-distance cutoff $2R$. If $p,q\in B(o,R)$ and $z$ is the midpoint of a minimizing geodesic from $p$ to $q$, then

$$d(p,q)\le 2R,d(o,z)\le d(o,p)+\frac{1}{2}d(p,q)\le 2R.$$

Up to the null cut relation and the harmless boundary case, this proves the inclusion

$$B(o,R)\times B(o,R)\subset {\mathcal{E}}_{B(o,2R)}(2R).$$

Moreover, for $0\le \ell \le 2R$,

$${\Omega }_{\ell /3}\subset B\left(o,\frac{8R}{3}\right),$$

whose volume is at most $(8/3{)}^n\mathrm{Vol}B(o,R)$ by Bishop—Gromov comparison. Writing $V(R)=\mathrm{Vol}B(o,R)$ and absorbing fixed numerical factors gives

$$V(R{)}^2\le C_{n,k}{\left(\frac{t}{\rho }\right)}^{k/2}{R}^{n-k}V(R).$$

After canceling the positive finite factor $V(R)$,

$$V(R)\le C_{n,k}{\left(\frac{t}{\rho }\right)}^{k/2}{R}^{n-k}.$$

Everything now reduces to proving that the curvature hypothesis supplies the required number of positive eigenvalues.

☘️ Unit-ball deficit and one direction

For $1\le k\le n$, let

$${\vartheta }_k(t,x)=\text{the sum of the k smallest eigenvalues of }{D}_t^W(x).$$

Near equality in the Bakry—Émery gradient estimate produces harmonic almost-splitting functions. Quantitatively, if ${\vartheta }_k(1,x_i)\to 0$ along a sequence of pointed manifolds with nonnegative Ricci curvature, then on every fixed ball one obtains $k$ harmonic functions whose gradients are almost orthonormal and whose Hessians have small averaged $L^2$ norm. In the rank-$n$ case, the ball is close to a Euclidean ball and its volume is close to Euclidean volume.

Proof mechanism: Heat frames from almost equality

Choose an orthonormal $k$-frame $e_i^1,\dots ,e_i^k$ realizing the smallest heat-deficit eigenvalues, and choose approximate maximizers ${\psi }_i^a$ in the variational definition of $G_1^W$. Put

$$U_i^a=\nabla P_1{\psi }_i^a(x_i),b_i^a=P_1|\nabla {\psi }_i^a{|}^2(x_i).$$

The identity

$$1-\left(2⟨U,e⟩-b\right)=|U-e{|}^2+b-|U{|}^2$$

decomposes the heat deficit into two nonnegative terms. The first makes the terminal vectors $U_i^a$ almost orthonormal. The second is the Bakry—Émery gap. With $u_s=P_{1-s}\psi$, the polarized Bochner formula gives

$$P_1|\nabla \psi {|}^2(x)-|\nabla P_1\psi (x){|}^2=2{\int }_0^1{P}_s\left(|\mathrm{Hess}{u}_s{|}^2+\mathrm{Ric}(\nabla u_s,\nabla u_s)\right)(x)ds.$$

Thus one can choose a common intermediate time at which the heat-averaged Hessian energies are small and the gradients form an approximate orthonormal frame. A Gaussian lower bound for the heat kernel turns these heat-weighted estimates into unweighted estimates on every fixed ball. Harmonic replacement preserves the approximate Gram matrix, and Bochner’s formula plus a cutoff gives small Hessian energy for the replacements. These are precisely the harmonic almost-splitting functions defined above.

Apply this with $k=n$. If no uniform trace gap existed at time $t=1/2$, there would be points $x_i$ such that

$$\mathrm{tr}{D}_{1/2}^W(x_i)⟶0.$$

Maximal-trace rigidity would then give

$$\mathrm{Vol}B(x_i,1)⟶{\omega }_n,$$

contradicting the assumed deficit ${\omega }_n-\epsilon$. Hence

$$\mathrm{tr}{D}_{1/2}^W(x)\ge \delta (n,\epsilon )>0$$

for every $x$. The largest eigenvalue is therefore at least $\rho =\delta /n$. The transport engine applies with $k=1$, fixed time $t=1/2$, and yields

$$\mathrm{Vol}B(o,R)\le C(n,\epsilon )R^{n-1}.$$

The constant is non-effective at this stage because the trace gap is obtained by compactness and contradiction.

☘️ Scalar curvature and two directions

Order the eigenvalues of $D_t^W$ as

$$d_1(t,x)\ge d_2(t,x)\ge \cdots \ge d_n(t,x)\ge 0,$$

and remove the largest one:

$$q_t(x)=\sum _{a=2}^n{d}_a(t,x)={\vartheta }_{n-1}(t,x).$$

Kong and Zhu prove that under

$$\mathrm{Ric}\ge 0,\mathrm{Scal}\ge \kappa >0,$$

there is a dimensional ${\delta }_n>0$ such that, at the curvature time $t={\kappa }^{-1}$,

$$q_t(x)\ge {\delta }_n.$$

Since $d_2$ is the largest of $d_2,\dots ,d_n$, this gives

$$d_2(t,x)\ge \frac{{\delta }_n}{n-1}=:{\rho }_n.$$

Thus at least two eigenvalues of $D_t^W$ are uniformly positive.

Proof: Excluding an $(n-1)$-fold almost splitting

If the gap failed after normalizing $\kappa =1$, then $q_1(x_i)\to 0$. Rank-$(n-1)$ heat rigidity would supply harmonic functions

$$h^1,\dots ,h^{n-1}$$

whose gradients are almost orthonormal and whose Hessians have small averaged energy on a large ball. Form the ordinary $(n-1)$-form

$$\beta =d{h}^1\wedge \cdots \wedge d{h}^{n-1}.$$

Almost orthonormality makes $|\beta {|}^2$ close to one on average. A cutoff Bochner argument makes both the Hessian energy and

$$\sum _{a=1}^{n-1}\mathrm{Ric}(\nabla h^a,\nabla h^a)$$

small on a sufficiently large interior ball. On $(n-1)$-forms, the curvature term in the Weitzenböck formula can be identified with the Ricci endomorphism through

$${\Lambda }^{n-1}{T}^{\ast }M\simeq T^{\ast }M\otimes {\Lambda }^n{T}^{\ast }M.$$

If $H_{ab}=⟨\nabla h^a,\nabla h^b⟩$, its quadratic form on $\beta$ is

$$⟨{\mathcal{R}}_{n-1}\beta ,\beta ⟩=\det (H)\mathrm{Scal}-\sum _{a,b}\mathrm{cof}(H{)}_{ab}\mathrm{Ric}(\nabla h^a,\nabla h^b).$$

The determinant is the squared norm $|\beta {|}^2$, while the cofactor matrix is $\det (H)H^{-1}$ wherever the gradients are independent. It records the Ricci trace over their $(n-1)$-plane. Under the canonical, orientation-free identification

$${\Lambda }^{n-1}{T}^{\ast }M\simeq T^{\ast }M\otimes {\Lambda }^n{T}^{\ast }M,$$

$\beta$ corresponds to a normal covector tensored with the determinant line. Locally choosing an orientation and writing that covector as $\eta =\ast \beta$, the displayed expression is

$$\mathrm{Ric}({\eta }^{♯},{\eta }^{♯}).$$

This makes its nonnegativity under $\mathrm{Ric}\ge 0$ transparent. When $H\approx I$, the scalar term is close to $\mathrm{Scal}$, whereas the cofactor-weighted tangential Ricci term is small by the preceding Bochner estimate. The scalar lower bound therefore forces the average of the expression to stay near one, whereas the integrated Weitzenböck and Bochner estimates make it of order $A^{-2}$ on a ball of radius $A$. Taking $A$ large gives a contradiction. This excludes $(n-1)$ almost-Euclidean directions even when the sequence collapses; no lower bound on unit-ball volume is required.

The common transport engine now applies with $k=2$, $t={\kappa }^{-1}$, and $\rho ={\rho }_n$. Therefore

$$\mathrm{Vol}B(o,R)\le C_n\left(\frac{{\kappa }^{-1}}{{\rho }_n}\right)R^{n-2}\le C_n{\kappa }^{-1}{R}^{n-2}.$$

This completes Kong—Zhu’s proof of the common codimension-two theorem.

🏷️ Proof III: Hodge Obstruction and Rank Improvement

Antonelli proves the stronger intermediate-curvature theorem without introducing a heat-induced metric. The quantity propagated through the proof is the $m$-volume ratio ${\mathcal{E}}_m(x,r)$ defined in the geometric preliminaries.

The index $m+1$ is forced by the desired exponent. A ball carrying only $m$ Euclidean directions is compatible with volume of order $r^m$; a ball carrying $m+1$ independent directions is the first rank that must be obstructed. The argument has two independent components. A Hodge estimate places an upper bound on the radius of any ball which almost splits ${\mathbb{R}}^{m+1}$. A critical-scale proposition then raises the splitting rank from $k$ to $k+1$ without decreasing ${\mathcal{E}}_m$ by more than a controlled factor. Starting from the vacuous rank $k=0$ and iterating $m+1$ times transports the original volume ratio to a terminal $(m+1)$-split ball, where curvature controls it (Antonelli, 2026).

☘️ Quantitative Hodge obstruction

The underlying tension can be stated before the estimates. An $(m+1)$-fold harmonic splitting produces one-forms which are nearly orthonormal and nearly covariantly constant, so their wedge has size close to one but very small derivative energy. Positive ${\mathcal{C}}_{m+1}$ forces that same collection of forms to carry definite Weitzenböck curvature energy. On a ball of radius $r$, the cutoff and Hessian errors cost only order $r^{-2}$, leading to the scale comparison ${\sigma }^2\lesssim r^{-2}$.

Suppose that $B_{4r}(x)$ carries harmonic functions

$$u=(u^1,\dots ,u^{m+1})$$

whose gradients are bounded and whose Gram matrix

$$G^{ab}=⟨d{u}^a,d{u}^b⟩$$

satisfies the scale-invariant splitting estimate

$$\frac{1}{\mathrm{Vol}{B}_{2r}(x)}{\int }_{B_{2r}(x)}\left(|G-I{|}^2+r^2\sum _{a=1}^{m+1}|\mathrm{Hess}{u}^a{|}^2\right)d\mathrm{Vol}\le \eta .$$

On the good set where $G$ is close to $I$, orthonormalize the one-forms by setting

$$\left(\begin{array}{c}{\beta }^1\\ ⋮\\ {\beta }^{m+1}\end{array}\right)=G^{-1/2}\left(\begin{array}{c}d{u}^1\\ ⋮\\ d{u}^{m+1}\end{array}\right),\Theta ={\beta }^1\wedge \cdots \wedge {\beta }^{m+1}.$$

Choose a cutoff $\chi =\chi (|G-I{|}^2)$ which is one when $G$ is very close to $I$ and vanishes before $G$ can become singular. Differentiating $G^{-1/2}$ and using

$$\nabla G^{ab}=\mathrm{Hess}{u}^a(\cdot ,\nabla u^b)+\mathrm{Hess}{u}^b(\cdot ,\nabla u^a)$$

gives

$$|\nabla \chi {|}^2+|\nabla \Theta {|}^2+\sum _{a=1}^{m+1}|\nabla {\beta }^a{|}^2\le C(n,m)\sum _{a=1}^{m+1}|\mathrm{Hess}{u}^a{|}^2.$$

Let $f$ be supported in $B_{2r}(x)$, equal to one on $B_r(x)$, and satisfy $|\nabla f|\le C_n{r}^{-1}$. The compactly supported forms

$$\tilde{\omega }=f\chi \Theta ,{\omega }^a=f\chi {\beta }^a$$

allow the Weitzenböck formula to be integrated without a boundary term.

Proof: From intermediate curvature to a bound on the splitting radius

For every compactly supported $p$-form $\xi$,

$${\int }_M⟨q_p(\mathrm{Rm})\xi ,\xi ⟩={\int }_M\left(|d\xi {|}^2+|\delta \xi {|}^2-|\nabla \xi {|}^2\right)\le C_n{\int }_M|\nabla \xi {|}^2.$$

Apply this once to $\tilde{\omega }$ and once to each ${\omega }^a$. On the support of $\chi$, the coframe is orthonormal, so the curvature identity from the preliminaries and ${\mathcal{C}}_{m+1}\ge {\sigma }^2$ give

$$⟨q_{m+1}(\mathrm{Rm})\tilde{\omega },\tilde{\omega }⟩+\sum _{a=1}^{m+1}⟨q_1(\mathrm{Rm}){\omega }^a,{\omega }^a⟩\ge 2{\sigma }^2{f}^2{\chi }^2.$$

The cutoff derivatives and the Hessian estimate therefore yield

$$2{\sigma }^2{\int }_{B_{2r}(x)}{f}^2{\chi }^2\le C(n,m)\left[r^{-2}\mathrm{Vol}{B}_{2r}(x)+{\int }_{B_{2r}(x)}\sum _{a=1}^{m+1}|\mathrm{Hess}{u}^a{|}^2\right]\le C(n,m)r^{-2}\mathrm{Vol}{B}_{2r}(x).$$

Markov’s inequality shows that the region where $|G-I|$ exceeds the cutoff threshold has volume at most $C\eta \mathrm{Vol}{B}_{2r}(x)$. If the scale-normalized negative Ricci bound is sufficiently small, Bishop—Gromov comparison gives

$$\mathrm{Vol}{B}_r(x)\ge c_n\mathrm{Vol}{B}_{2r}(x).$$

Choosing $\eta$ small ensures that $f=\chi =1$ on a subset of $B_r(x)$ with volume at least $c_n\mathrm{Vol}{B}_{2r}(x)$. Hence

$${\sigma }^2\mathrm{Vol}{B}_{2r}(x)\le C(n,m)r^{-2}\mathrm{Vol}{B}_{2r}(x).$$

Canceling the positive volume proves

$$\sigma r\le C(n,m).$$

Both sides of the last estimate are dimensionless. It says exactly that an $(m+1)$-fold almost splitting cannot persist beyond a dimensional multiple of the curvature length ${\sigma }^{-1}$.

Cheeger—Colding harmonic replacement now turns metric almost splitting into the harmonic hypotheses above. Consequently, for dimensional constants ${\epsilon }_H,C>0$,

$$\begin{array}{rl}& \mathrm{Ric}\ge -{\epsilon }_H{r}^{-2}g,{\mathcal{C}}_{m+1}\ge {\sigma }^2,\\ & B_r(x)\text{ is }(m+1,{\epsilon }_H)\text{-split}\end{array}⟹r\le C(n,m){\sigma }^{-1}.$$

Thus positive $(m+1)$-intermediate curvature forbids $m+1$ almost-Euclidean directions above the curvature scale.

☘️ Critical scales and rank improvement

The second component is independent of the intermediate-curvature hypothesis. For every $k\le m$ and target accuracy $\epsilon$, sufficiently accurate $k$-splitting implies the existence of a possibly smaller ball with one additional Euclidean direction:

Rank-improvement proposition

There are $\delta ,c>0$, depending only on $n,m,k,\epsilon$, such that if

$$\mathrm{Ric}\ge -\delta r^{-2}g$$

and $B_r(p)$ is $(k,\delta )$-split, then there are $q\in M$ and $0<s\le r$ for which $B_s(q)$ is $(k+1,\epsilon )$-split and

$${\mathcal{E}}_m(q,s)\ge c{\mathcal{E}}_m(p,r).$$

The need to preserve ${\mathcal{E}}_m$, rather than merely finding a better-split ball, is the reason for the critical-scale construction. Its structure comes from the rescaling method of Kapovitch and Wilking (Kapovitch & Wilking, 2011).

Proof mechanism: Selection of a volume-heavy critical cell

Rescale to $r=1$. The splitting assumption identifies a large neighborhood of $p$ with a small-error approximation to

$${\mathbb{R}}^k\times Z.$$

If the residual factor $Z$ contains a sufficiently long minimizing segment through the basepoint, recenter at the segment midpoint. The quantitative almost-splitting theorem turns that segment into another Euclidean direction on a fixed smaller ball. Bishop—Gromov comparison transfers a definite fraction of ${\mathcal{E}}_m(p,1)$ to this ball.

The harder case is that $Z$ is macroscopically small. The original ball is then close to ${\mathbb{R}}^k$, and harmonic replacement supplies a nearly Euclidean projection

$$b:B_2(p)⟶{\mathbb{R}}^k.$$

For a good point $q$, measure the failure of $b$ to preserve distances on $B_s(q)$ by

$$D_{q,b}(s)=\underset{x,y\in {\overline{B}}_s(q)}{\sup }\left|d(x,y)-|b(x)-b(y)|\right|.$$

Choose the largest critical radius $t(q)$ at which

$$D_{q,b}(t(q))=c_{\ast }t(q)$$

for a fixed small $c_{\ast }(n)>0$. A maximal-function estimate makes the set of points with uniform harmonic-splitting control occupy a definite fraction of $B_1(p)$. A Vitali selection in the Euclidean $b$-image produces disjoint cells $E_j$, centers $q_j$, and critical radii $t_j\le 1$ satisfying

$$\mathrm{Vol}\left(\underset{j}{⨆}{E}_j\right)\ge c_n\mathrm{Vol}{B}_1(p),E_j\subset B_{C{t}_j}(q_j),\sum _j{t}_j^k\le C(n,k).$$

Because $k\le m$ and $t_j\le 1$,

$$\sum _j{t}_j^m\le \sum _j{t}_j^k\le C.$$

At least one cell is therefore heavy at the $m$-dimensional normalization:

$$\frac{\mathrm{Vol}(E_j)}{t_j^m}\ge \frac{\mathrm{Vol}({⨆}_{\ell }{E}_{\ell })}{{\sum }_{\ell }{t}_{\ell }^m}\ge c\mathrm{Vol}{B}_1(p)=c{\mathcal{E}}_m(p,1).$$

At the critical scale, a blow-up is close to ${\mathbb{R}}^k\times Y_j$, and the defining distance distortion ensures that $Y_j$ is not a point: it contains a segment of length comparable to $t_j$. Recenter near the midpoint of that segment and apply quantitative almost splitting. This gives a $(k+1,\epsilon )$-split ball of radius $s=\vartheta t_j$. Finally, $E_j\subset B_{C{t}_j}(q_j)$ and Bishop—Gromov comparison give

$${\mathcal{E}}_m(q,s)\ge c\frac{\mathrm{Vol}(E_j)}{t_j^m}\ge c{\mathcal{E}}_m(p,1).$$

This proves rank improvement in both residual-factor cases.

The exponent restriction $k\le m$ enters exactly in the inequality ${\sum }_j{t}_j^m\le {\sum }_j{t}_j^k$. It is what lets the induction gain directions while retaining the $m$-dimensional volume ratio.

☘️ Iteration and the final volume bound

Fix the terminal accuracy ${\epsilon }_{m+1}={\epsilon }_H$ from the Hodge obstruction. Choose the preceding accuracies

$${\epsilon }_m,{\epsilon }_{m-1},\dots ,{\epsilon }_0$$

backward so that rank improvement at accuracy ${\epsilon }_k$ produces accuracy ${\epsilon }_{k+1}$. Every ball is trivially $(0,{\epsilon }_0)$-split. Starting from $B_R(p)$, successive applications give

$$R=r_0\ge r_1\ge \cdots \ge r_{m+1}>0$$

and centers $p_k$ such that $B_{r_k}(p_k)$ is $(k,{\epsilon }_k)$-split and

$${\mathcal{E}}_m(p_{k+1},r_{k+1})\ge c_k{\mathcal{E}}_m(p_k,r_k)(0\le k\le m).$$

If the global Ricci bound is $\mathrm{Ric}\ge -{\delta }^{2}g$ and $\delta R\le \nu (n,m)$ for a sufficiently small $\nu$, then

$${\delta }^2{r}_k^2\le {\delta }^2{R}^2\le \nu (n,m{)}^2\le {\epsilon }_k,$$

so every rank-improvement step has the required scale-normalized Ricci bound.

The product $\delta R$ is dimensionless. Because every selected radius satisfies $r_k\le R$, smallness at the initial scale automatically supplies the almost-nonnegative Ricci hypothesis at every later scale; when $\delta =0$, there is no restriction on $R$.

The terminal ball is $(m+1,{\epsilon }_H)$-split. The Hodge obstruction gives

$$r_{m+1}\le C(n,m){\sigma }^{-1}.$$

Bishop comparison under $\delta r_{m+1}\le \delta R\le \nu \le 1$ gives

$$\mathrm{Vol}{B}_{r_{m+1}}(p_{m+1})\le C(n,m)r_{m+1}^n.$$

Consequently,

$${\mathcal{E}}_m(p_{m+1},r_{m+1})\le C(n,m)r_{m+1}^{n-m}\le C(n,m){\sigma }^{-(n-m)}.$$

Reversing the chain of rank-improvement inequalities yields

$$\frac{\mathrm{Vol}{B}_R(p)}{R^m}={\mathcal{E}}_m(p_0,r_0)\le \left(\prod _{k=0}^m{c}_k^{-1}\right){\mathcal{E}}_m(p_{m+1},r_{m+1})\le C(n,m){\sigma }^{-(n-m)}.$$

This is Antonelli’s general theorem. In the scalar-curvature case, take

$$m=n-2,{\mathcal{C}}_{n-1}=\frac{1}{2}\mathrm{Scal},{\sigma }^2=\frac{\kappa }{2}.$$

Then ${\sigma }^{-(n-m)}={\sigma }^{-2}=2{\kappa }^{-1}$, and the common codimension-two estimate follows after absorbing the factor $2$ into $C_n$.

🏷️ Comparison of the Proof Mechanisms

The three approaches agree on the geometric interpretation but use different detectors and different mechanisms for converting missing directions into a volume estimate.

• Ge measures the total Fisher trace loss and proves that its source contains two units after an additional heat average. The proof is scalar after the weighted eigenline estimate: two parabolic source equations force entropy decay.

• Kong—Zhu retain directional information in the eigenvalues of $D_t^W$. Splitting rigidity supplies one or two positive directions, and angular coercivity converts their number directly into a power of $L^{-1}$ for long minimizing geodesics. Their codimension-two result is one proof of the common theorem; the one-direction case is the additional codimension-one theorem.

• Antonelli does not build a heat metric. The Hodge identity directly excludes $(m+1)$ harmonic splitting directions above the curvature scale, while critical-scale rank improvement searches for one new direction at a time and preserves the normalized volume ratio.

• Differential forms enter all three arguments in distinct ways. Ge’s second production unit uses a weighted lowest-eigenline estimate. Kong—Zhu exclude an $(n-1)$-fold splitting with an $(n-1)$-form. Antonelli sums the Weitzenböck terms of one $(m+1)$-form and its $m+1$ constituent one-forms to recover exactly $2{\mathcal{C}}_{m+1}$.

• The volume-conversion stages are genuinely different. Ge turns two source units into $S(t)\lesssim -\log t$ and then a $t^{-1}$ relative volume loss at radius $\sqrt{t}$. Kong—Zhu turn $k$ defective directions into $L^{-k}$ scarcity of long geodesics. Antonelli bounds the terminal $(m+1)$-split scale and transports its $m$-volume ratio backward through $m+1$ rank improvements.

• Antonelli’s theorem has the broadest curvature statement: it treats every ${\mathcal{C}}_{m+1}$ and an almost nonnegative Ricci lower bound on scales satisfying $\delta R\le \nu (n,m)$. Ge and Kong—Zhu work under globally nonnegative Ricci curvature, but their heat methods give direct all-scale scalar-curvature estimates.

Scope of the conclusion

The common conclusion is a volume-growth theorem; an $R^{n-k}$ volume bound alone does not construct a uniformly cobounded map to an $(n-k)$-complex. Antonelli additionally combines his general volume bound with a separate covering theorem to obtain bounded $m$-dimensional Urysohn width under the noncollapsing hypothesis

$$\underset{p\in M}{\inf }\mathrm{Vol}{B}_1(p)\ge v_0$$

and a sufficiently small negative Ricci lower bound. That additional conclusion uses noncollapsing and is not a formal consequence of the volume estimate by itself (Antonelli, 2026).

🏷️ Hypotheses, Constants, and Non-implications

Several distinctions are essential to the precise statements.

• Ge and Kong—Zhu assume $\mathrm{Ric}\ge 0$. Antonelli permits $\mathrm{Ric}\ge -{\delta }^{2}g$, but only obtains the estimate at radii satisfying $\delta R\le \nu (n,m)$. Positive scalar curvature alone is not a hypothesis of any of the three theorems.

• The codimension-one hypothesis is uniform in the center:

  $$\underset{x\in M}{\sup }\mathrm{Vol}B(x,1)\le {\omega }_n-\epsilon .$$

  A deficit at one selected point is not the theorem’s assumption. The radius $1$ is only a normalization of the macroscopic scale.

• In the common codimension-two theorem, $\kappa$ denotes the scalar-curvature lower bound and has units of ${\mathrm{length}}^{-2}$. The factor ${\kappa }^{-1}$ has units of ${\mathrm{length}}^2$ and exactly compensates for the two missing powers of $R$. In Antonelli’s notation, ${\mathcal{C}}_{m+1}\ge {\sigma }^2$, so $\sigma$ has units of ${\mathrm{length}}^{-1}$.

• Every estimate is uniform in the ball center. Ge and Kong—Zhu obtain the scalar-curvature estimate at all radii. Antonelli does so when $\delta =0$; for $\delta >0$, the theorem is restricted to $\delta R\le \nu (n,m)$.

• None of the three papers classifies equality or almost-equality cases. The products $S_a^1\times {\mathbb{R}}^{n-1}$ and $S_a^2\times {\mathbb{R}}^{n-2}$ establish sharpness of the growth exponents, not uniqueness of sharp models.

• The codimension-one constant $C(n,\epsilon )$ is non-effective in the Kong—Zhu proof because its trace gap is obtained by compactness. The codimension-two statements and Antonelli’s intermediate-curvature theorem provide controlled constants but do not optimize their numerical values.

• Volume-growth codimension is weaker than a bound on Urysohn width or macroscopic dimension. Antonelli’s noncollapsed Urysohn-width corollary requires extra input, as recorded above.

🔗 See Also

• on curvature-dimension geometry of metric measure spaces — The $\mathrm{CD}(0,N)$ viewpoint explains why nonnegative Ricci curvature simultaneously controls Wasserstein transport, heat-flow contraction, Bishop—Gromov comparison, and stability under measured limits.
• on convergence of graph Laplacian to manifold’s Laplacian — Both notes treat diffusion as a probe of latent geometry; here the heat kernel detects the number of macroscopic directions rather than convergence of a discrete operator.
• on concentration of information and log-concave distributions — The Fisher-information and entropy components of Ge’s proof are geometric counterparts of information-content methods for probability densities.

📚 References

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