Gigli’s distributional curvature characterization of Alexandrov spaces

1. From (A) to (B)

The external Alexandrov-to-RCD theorem supplies unreduced $\mathrm{RCD}((n-1)\kappa,n)$, without restricting to spaces without boundary. The additional content of Theorem 3.1 is the curvature inequality for all $X,Y,f$, obtained using weak derivatives alone.

[KKS23, Theorem 2.9; Corollary 2.10, pp. 7–8]

2. From (B) to (A)

Theorem 4.1 selects the same calibrated geodesic from each terminal point when the terminal weight changes. This permits the pointwise countable infimum of trial energies in Theorem 5.10. Theorem 6.3 then obtains both comparison along every geodesic and $\dim_HM=n$. The companion theorem supplies the passage from almost-everywhere path information to an arbitrary prescribed geodesic.

§7, proof of Theorem 1.1 · p. 50

1. Geometric and analytic bounds for the same energy difference

Choose gradients $A,B$ of compact test functions with bounded Laplacian independently from a finite sign-symmetric law. Use short flows with velocity $hA+h^2tS_A$. Write $J_{0A}^h$ for their radial action and $J_{AB}^h$ for the optimal transport action between endpoint measures. Alexandrov cosine comparison gives

For the other bound, the weak $L^2$ first variation of interpolation densities and weighted strong convergence of velocities permit an expansion with Sobolev dual potentials. Sign symmetry removes odd terms. First send $h\downarrow0$; then let $S_A\to-\nabla_AA$ to remove the radial acceleration defect. The remaining difference is controlled by curvature and a nonnegative square:

This is an integrated weak-calculus argument. Pairing the first density variation with fixed $L^2$ Hessian coefficients is what justifies the fourth-order term.

Lemmas 3.3–3.6; Proposition 3.7 · pp. 12–19

2. Local gradient choices remove the square

A general $L^2$ vector field need not be approximable by one gradient, but it can be approximated using separate gradients on a finite partition. Apply Lemma 3.9 to $-\nabla_A(B-A)$ and split the weight into $f\alpha_\ell$. Linearity in the weight and $\sum\alpha_\ell=1$ recombine even the differentiated-weight terms exactly, so no uniform bound on the partition derivatives is needed. The square error tends to zero. For the uniform law on $\{U,-U,V,-V\}$, only mixed pairs remain, giving the sectional inequality.

Lemmas 3.8–3.9; Proposition 3.10 · pp. 19–21

3. Globalization comes last

For a fixed finite gradient family, constant-coefficient nonnegative curvature functionals are positive Radon measures. Positivity of the resulting coefficient polynomial on rational vectors extends the inequality to variable test coefficients. Local approximation by potentials with bounded Laplacian then gives strong $L^2$ convergence of the fields, their divergences, and their covariant derivatives, which passes the curvature functional to the limit. Finally exhaust the support of the weight using $f\chi_R$. The original tensor integrands are globally integrable, and terms involving $d\chi_R$ are controlled by $L^1$ tails. This order recovers the original global quantifiers of Theorem 1.1.

Lemmas 3.11–3.12; Remark 3.13 · pp. 21–23

[Gig19, Definition 2.6; Proposition 2.7, pp. 5–6]

1. Fix the terminal density and control the singular Bochner term

First take $f=f_0+\lambda e^{-Dd(o,\cdot)}>0$. For viscosity $c>0$, construct $H_t^c=P_{ct}(e^{-u/(2c)})$, $J_t^c=P_{c(1-t)}(f/H_1^c)$, and $\rho_t^c=H_t^cJ_t^c$. Put $\varphi=-2c\log H$, $\psi=2c\log J$, $\theta=(\psi+\varphi)/2$, and $b=c\log\rho$. Then $v^c=\nabla\theta$ is the continuity-equation velocity and $w^c=\nabla b$ the density correction.

Applying (B) in the Riccati identity and completing a square gives the index integrand and the acceleration error $H_{|w^c|^2/2+c\Delta b^c}$. Smallness of $w^c$ alone cannot handle the $1/c$ terms introduced by differentiating the density. The manuscript cancels these terms exactly before decomposing the remaining Bochner measure into a nonnegative measure and an $L^2$ remainder. The bound $c\Delta b^c\ge-e_c(t)$ with $\|e_c\|_{L^1(\delta,1)}\to0$, applied to quasi-continuous representatives, controls the singular measure contribution uniformly as the terminal cutoff $T\uparrow1$.

Lemmas 4.5–4.8; Proposition 4.9 · pp. 28–33

[JLZ16, Theorem 1.2, p. 2]

2. Equality of actions justifies the quadratic-energy limit

Convergence $\varphi_t^c\to Q_tu$ and $W_2$ compactness of the density curves alone do not pass the index form to the limit. The action identity and Hopf–Lax duality force equality throughout the intervening estimates. This simultaneously identifies the limit as a Wasserstein geodesic and gives

Truncation extends this convergence to multiplication by the fixed $L^2$ coefficient $\nabla Y$. Uniqueness of optimal transport viewed backward from the absolutely continuous terminal measure gives the same path prescription under changes of terminal weight. The limits are $T\uparrow1$, then $c\downarrow0$, then $\lambda\downarrow0$. Bounded compression is never assumed at the point-source time zero.

Lemmas 4.10–4.13; proof of Theorem 4.1 · pp. 33–37

[GH15, Theorem 3.5, p. 11] [GRS16, Theorems 1.1, 1.3, p. 2]

1. Upgrade spatial a.e. frames to frames at every path time

Noncollapse, $m=\mathcal H^n$, gives $n$-dimensional Euclidean tangents almost everywhere. A countable selection of harmonic splitting maps and an energy estimate for Gram-matrix rank defects split the exceptional set into a set $N$ of zero 2-capacity and an $\mathcal H^{n-1}$-null set $S$. Almost every entire path of a bounded-speed, bounded-compression plan avoids both. Deng’s Hölder continuity of tangent cones from the same rescaling sequence extends regularity from a dense set of times to all interior times. Compression handles the endpoints.

Positive minors of the continuous Gram matrices give local time-dependent frames. Solving finite-dimensional ODEs with $L^2$ connection coefficients constructs parallel frames. Gigli–Tamanini’s second-order differentiation formula makes the path velocity parallel. These constructions use the noncollapsed RCD hypotheses; the sectional bound enters in the radial application below.

Proposition 5.1; Lemmas 5.4–5.5; Proposition 5.6 · pp. 38–44

[DPG18, Theorem 1.12; Corollary 2.14, pp. 7, 17]

[BPS21, Proposition 1.4, p. 5] [Deng25, Theorems 1.1–1.2, p. 2] [GT21, Theorem 5.13, p. 50]

2. The countable infimum and the Jacobi trial

For $u=\min\{2R_\kappa r,2R_\kappa^2\}$, $Q_1u=q$ on the ball and $p$ is the unique minimizer. The calibrated paths are radial geodesics from $p$. An $L^2$ terminal difference-quotient estimate identifies $v(1)=\nabla q$.

One fixed trial gives $E_0m-H_q(Z,Z)\ge0$. Riesz representation yields $H_q(Z,Z)=h_Zm+\mu_s$ with $\mu_s\le0$. A countable trial family satisfying global endpoint identities then gives $h_Z(x)\le\inf_Y E_Y(x)$ outside one common null set. Along each path, let $P_\parallel,P_\perp$ be the parallel translates of $Z_\parallel,Z_\perp$. The field

is approximated by the countable family in the pathwise $W^{1,2}$ norm. Its index energy equals

A countable sequence $\delta\downarrow0$ gives the stated measure inequality. No measurable selection of the pathwise approximating sequences, and no density in an integrated space of fields along the whole plan, is required.

Lemmas 5.7–5.9; Theorem 5.10 · pp. 44–47

1. Construct bounded functions to which the companion theorem applies

Put $m_\kappa(r)=\int_0^r s_\kappa(s)\,ds$. Fix $0<a<b<R_\kappa$ and a smooth nonincreasing $\theta$ equal to one on $[0,a^2/2]$ and zero on $[b^2/2,\infty)$. Define $\chi(0)=0$, $\chi'(q)=\theta(q)s_\kappa(\sqrt{2q})/\sqrt{2q}$, and $F_p=\chi(r^2/2)$. Then $F_p$ is bounded globally Lipschitz and constant outside the ball. In the Hessian chain rule, the additional radial term $r\theta's_\kappa$ is nonpositive, and the singular part retains its sign. Thus the bounded continuous function $G_p=\theta(r^2/2)s_\kappa'(r)$ satisfies $H_{F_p}(Z,Z)(h)\le\int hG_p|Z|^2dm$ for every compact test gradient $Z$ and nonnegative compact Lipschitz weight $h$. On $B_a(p)$, $F_p=m_\kappa(r)$ and $G_p=1-\kappa F_p$.

For a full-support $\mathrm{RCD}(K,N)$ space with $1<N<\infty$, the companion theorem turns precisely these boundedness, continuity, and weak-Hessian hypotheses into $(F_p\circ\sigma)''\le\ell^2G_p\circ\sigma$ along every constant-speed minimizing geodesic $\sigma$ of length $\ell$, in distributions on $(0,1)$. Neither $F_p\in L^2(m)$ nor an $L^2$ Hessian is required. The reference measure in this application remains unchanged.

Theorem 6.1; Lemma 6.2 · pp. 48–49

[OpenAI26b, Theorem 1.1, p. 2]

2. The model ODE, local four-point comparison, globalization, and dimension

For a triangle $[pxy]$ of perimeter less than $a$, its chosen side $\sigma$ lies in $B_a(p)$. Hence $y=m_\kappa(d(p,\sigma_t))$ satisfies $y''+\kappa\ell^2y\le\ell^2$. The corresponding function $\bar y$ on the model triangle solves the equality with the same endpoint values. Testing with the negative part of $y-\bar y$ gives $y\ge\bar y$ directly when $\kappa\le0$, and by the Dirichlet Poincaré inequality and $\kappa\ell^2<\pi^2$ when $\kappa>0$. Monotonicity of $m_\kappa$ recovers point-on-side distance comparison.

Converting these short-triangle comparisons to four-point comparison in a small ball uses ambient minimizing segments; convexity of the ball is not assumed. The local CBB criterion and globalization for complete length spaces give $\mathrm{CBB}(\kappa)$. Finally, every nonempty bounded ball has positive finite $\mathcal H^n$ measure. A countable exhaustion gives $\dim_HM=n$, completing both the curvature and dimension conclusions of (B)$\Rightarrow$(A).

Theorem 6.3 · pp. 49–50

[AKP24, Theorems 8.14, 8.30, 8.31, pp. 86, 93–95]