Gigli’s distributional curvature characterization of Alexandrov spaces

Proof. The good-cutoff construction and the integrated Bochner estimate give (i) and (2.3); see [19, Appendix A, Lemma A.2 and Equation (A.13)]. For the local version of the cutoff construction one uses finitely many small good cutoffs inside U, and a smooth truncation of their sum. This keeps the Laplacian bounded and Sobolev by the test algebra and chain rules.

Let P_t denote the heat semigroup. For h\in L^2\cap L^\infty, positive-time regularization gives P_th\in\Test^\infty [19, Appendix A, Equation (A.7)]. Spectral calculus and density of bounded L^2 data show that these regularizations form a core. For each such regularization h,

The uniform cutoff bounds and the L^2 tails imply convergence to \Delta h in L^2, as well as convergence of the functions and gradients. This proves the compact-core assertion.

For (iii), choose nested cutoffs \chi,\zeta such that \chi=1 near the region in question and \zeta=1 near \supp\chi. Put

The function \zeta g lies in \Test. The product rule, spectral continuity and the fixed cutoff bounds give convergence of g_t to \chi\zeta g in the Laplacian graph norm. The Bakry gradient estimate gives

Apply (2.3) to g_t-\chi\zeta g to obtain strong Hessian convergence. Since \chi\zeta g=g near the region, (iii) follows.

For (iv), use \xi_t=\chi P_t\xi, with an outer cutoff \chi=1 near \supp\xi. The heat semigroup converges strongly in W^{1,2}; the uniform Lipschitz estimate and the small-time heat-kernel moment bound give uniform convergence on the common compact support. The cutoff product gives the same conclusions for \xi_t, and positivity is preserved.□

Proof. Expand \nabla_X(fW)=X(f)W+f\nabla_XW. Every coefficient of f is a product of two L^2 factors and bounded factors. The coefficients of \nabla f lie in L^2. Uniform convergence of f_j pairs with the fixed L^1 coefficient, and weak L^2 convergence of their gradients pairs with the fixed L^2 coefficient. This proves the weight-continuity assertions. Approximation then extends the identities.

For the field assertion,

The first term tends to zero in L^2 by uniform boundedness of X_j; the second does so by dominated convergence. The brackets therefore converge in L^2 as well. Products of the resulting strongly convergent L^2 quantities converge in L^1. Any remaining uniformly bounded field factor is handled by almost-everywhere convergence and domination. This includes the last term of ().□

Proof. The fields in (3.3) are bounded, have L^2 covariant derivative and bounded divergence, and have support in a fixed compact set. The Sobolev flow theorem applies [5, Theorems 4.3, 5.4, 6.7 and 8.3; Section 9.6]. Applying it also to the time-reversed field, and using uniqueness, gives the inverse flow. The divergence bound is O(h), so both compression constants are 1+O(h). The speed is bounded by h\|A\|_\infty+h^2\|S_A\|_\infty, which gives the displacement and support assertions. Properness is used here to replace bounded closed neighborhoods by compact sets.

To check (3.4), the pushforward of m under the flow has density between 1-O(h) and 1+O(h), by the forward and inverse compression bounds. Replacing f(z) by f(F^h_{r,A}(z)) in the pushforward changes the density by at most O(h)\operatorname{Lip}(f), again by compression and the displacement bound. Multiplication of the pushed-forward reference density by f changes it from f by at most O(h)\|f\|_\infty. This proves (3.4) without requiring a spatially Lipschitz velocity field.

The continuity equation, tested with q\in W^{1,2}, reads

All pairings are finite because the densities are bounded and have bounded support. Divide by h and use (3.4). The right-hand side converges, uniformly in r, to r\int f\langle\nabla q,A\rangle=r\int qL_A. The uniform L^\infty bound on the left quotient, followed by density of compact Lipschitz functions in L^1 on bounded sets, proves (3.5), also in space-time. Use (3.8) once more, with r=1, to obtain

The functions paired with the normalized density difference are fixed and integrable on its support. Equation (3.5) therefore gives (3.6).

For completeness, if v\in L^1 on a common bounded neighborhood, then

Indeed this holds for continuous functions by the displacement bound, and follows for L^1 functions by approximation and the uniform compression bound. Differentiation of a Lipschitz function along the flow gives

The preceding composition convergence proves (3.7). A cutoff may be inserted when g is only locally Lipschitz.□

Proof. At m-almost every base point z, the Alexandrov tangent cone is Euclidean. Choose a geodesic from z to each endpoint F^h_{1,A}(z). For the finite weights p_A, the radial vectors v_A\in T_zM satisfy

Actual angles are at least the corresponding \kappa-comparison angles. Consequently,

Zero radial lengths contribute zero. All sides are O(h), so the model triangles are defined for sufficiently small h, including when \kappa>0.

For radial side lengths a,b and opposite side length e, put z_0=(a^2+b^2-e^2)/2. The model cosine law gives, uniformly for such small triangles,

For \kappa=0 this is exact without the correction. For \kappa\ne0, substitute the Taylor series of the model sine and cosine into their cosine law. After multiplication by ab, the denominators are the nonvanishing analytic factors \mathfrak s_\kappa(a)/a and \mathfrak s_\kappa(b)/b, where \mathfrak s_\kappa''+\kappa\mathfrak s_\kappa=0, \mathfrak s_\kappa(0)=0, and \mathfrak s_\kappa'(0)=1. This also proves uniformity at degenerate triangles and zero sides.

We first identify the leading distances. By integration of the speed and the composition convergence used in Lemma 3.2,

There is a countable collection of scalar 1-Lipschitz functions whose differentials norm every tangent vector almost everywhere; one may use the norming family in [5, Lemma 9.2], with local cutoffs when needed. For each member g, (3.7) and its difference for two flows give the lower bounds

First taking a finite part of the norming family, and then increasing it, shows that the negative parts of a^h_A/h-|A| and e^h_{AB}/h-|A-B| tend to zero in measure. The radial convergence follows. All scaled distances are uniformly bounded, so this convergence also holds in every finite L^p(fm).

Equations (3.11)–(3.12), initially retaining only their quadratic terms, imply

By (3.2), the last quantity is \mathbb E\int f|A-B|^2. The pair-distance lower bounds already proved imply convergence of the negative errors in L^1. The displayed upper bound therefore forces the positive errors to converge in L^1 as well. Since the family is finite, this proves the pair convergence in (3.9) for every pair of positive probability.

We may now retain the quartic term in (3.12). The convergence just proved gives

in L^1(fm). Thus

The radial energy dominates half the squared radial displacement, and (F^h_{1,A},F^h_{1,B})_\#(fm) is a coupling of the endpoint measures. Hence

Their combination with (3.13) proves (3.10).□

Proof. Restriction of the plan. The endpoints have uniformly bounded densities and support in a common ball. All connecting geodesics lie in a larger fixed ball, which is compact. The standard interpolation bound gives the uniform density estimate. We record a sharper consequence needed below. Weaken the Ricci parameter to a fixed K'<0. The zero-mass case is immediate; otherwise let \pi^h be the optimal dynamical plan and normalize it to a probability \pi, with marginals r_t m. By [18, Theorem 1.1], \pi has a Borel conull set on which every evaluation e_t, t<1, is injective. Both endpoints are absolutely continuous. Applying the same theorem to the reversed plan, pulling its conull set back under time reversal, and intersecting with the first set gives a Borel conull \Gamma on which e_t is injective for every t\in[0,1].

Fix s\in(0,1), and let H\geq0 be bounded Borel on \Gamma, with Z=\int H\dd\pi>0. The probability \pi_H=(H/Z)\pi is optimal for its endpoint marginals: after scaling H\leq1, a cheaper transport for H\pi could be spliced with \pi-H\pi, contradicting optimality. Its endpoint densities are bounded and absolutely continuous. The integrated \CD(K',n) inequality supplies an optimal dynamical plan for these marginals, and uniqueness forces that plan to be \pi_H. The measurable inverses of e_t|_\Gamma give, by Lusin–Souslin,

where h_t is extended by zero off e_t\Gamma. Write \beta_t(\ell)=\tau_{K',n}^{(t)}(\ell) for the unreduced distortion coefficient and set

The integrated inequality for \pi_H, after cancellation of Z^{1/n-1}, is exactly

There are no undefined inverse powers in this calculation: zero or infinite density values lie over marginal-null sets at the three fixed times, and on the common finite-measure compact support Q,

Taking H to be indicators of Borel sets proves F_s\geq0 \pi-almost everywhere. Restoring the common mass preserves this homogeneous inequality. Its exceptional set may depend on s; every fixed-time estimate below has the same constants, which suffices for the supremum over s. Jointly Borel density versions and Fubini give the corresponding \dd s\otimes\pi-almost-everywhere statements for time integrals.

The density norm bound. On the bounded range of possible geodesic lengths \ell, the pointwise CD distortion coefficients, divided by their zero-curvature coefficients, have a positive lower bound and are at least 1-C\ell^2, uniformly in s. The negative power mean is at most the arithmetic mean. Since the endpoint densities are bounded, the pointwise CD inequality therefore implies, along the optimal dynamical plan \pi^h,

To see that the error remains valid when \ell is not small, let a(\ell) be a common positive lower bound for the two distortion ratios. The CD inequality first gives the negative power mean multiplied by a(\ell)^{-n}. On the fixed length interval, a(\ell)^{-n}-1\leq C\ell^2. This proves (3.16), including such lengths.

Integration against \pi^h gives

Semiconcavity of f gives

Here \int\ell^2\dd\pi^h=2J^h_{AB}=O(h^2), by the configuration coupling. Adding the last two inequalities and \int f^2 proves the norm bound in (3.14), using (3.4) for the endpoints.

First variations of velocity and density. The same coupling and (3.9) imply

The continuity equation tested with u_B-u_A, and then (3.6), gives

The norm estimate for the densities also gives \int_0^1\int\sigma^h_s|D|^2\dd s\to\int f|D|^2. Expanding the square proves (3.15).

Finally, for any fixed q\in W^{1,2}(M),

The first term converges to \int qL_A. In the second, use (3.15) and Cauchy–Schwarz to replace v^h_t/h by D; then use the density norm bound to replace \sigma^h_t by f. The error tends to zero uniformly in s. The limit is s\int f\langle\nabla q,D\rangle =s\int qL_D. Boundedness of the normalized density differences in L^2(\dd s\dd m), and density of finite sums of separated space-time test functions, prove the weak convergence in (3.14).□

Proof. The energy identity. For a bounded-compression solution \sigma_s of the continuity equation with velocity v_s, use

Its strong W^{1,2} dependence on time and the continuity equation give the exact energy identity

One may verify the time differentiation directly by writing p_s as a sum of fixed Sobolev functions with affine time coefficients. The endpoint terms are finite by the bounded densities and bounded support; the remaining terms are finite by the kinetic-energy and W^{1,2} bounds. Thus no boundedness assumption on q_{AB} is needed.

The dual expansion. The term -s|D|^2/2 in q_s cancels the quadratic part of the Hamilton–Jacobi residual. Thus the residual begins at order h^3, and the first density variation supplies its contribution at order h^4. Explicitly,

For the optimal interpolation, Lemma 3.4 gives

The first limit uses weak L^2 convergence against the fixed L^2 function \langle D,Q_s\rangle. The second uses convergence in measure and the uniform density bound, by truncation of the fixed integrable function |Q_s|^2. Outside the common density support both statements are vacuous.

Apply (3.6) to the two fixed endpoint functions q_0,q_1. Leave the endpoint contribution involving u_B-u_A unexpanded. The terms of order four are then precisely (3.18). For the cubic term, integration by parts gives

Using Q_1=Q_0-T_D and \int_0^1Q_s\dd s=Q_0-T_D/2, its difference from the cubic residual is

Also q_1-q_0=-|D|^2/2. Dropping the nonnegative last term in (3.20) proves (3.17).

The radial defect. For the radial flow use its actual velocity from (3.3), and use Lemma 3.2 in place of Lemma 3.4. Here D=A, C_s=sA, and Q_s=-s\nabla_AA. The last term of (3.20), divided by h^4, is

The integrand in the spatial limit is fixed and integrable, and (3.4) justifies the limit. This proves the radial assertion.□

Proof. Integration by parts. We first justify the identity

For test q, integration by parts in its two terms gives

The Hessian terms cancel. The vector fields paired with \nabla q on both sides of (3.26) are in L^2 and have bounded support. Density of test functions in W^{1,2} therefore proves the identity for the stated class, without assigning a Hessian to q.

Square completion. The maps s\mapsto F_{C_s} and s\mapsto Q_s are polynomials with values in L^2. They have derivatives

Express the endpoint difference in K_{AB} as the integral of the derivative of \langle F_{C_s},Q_s\rangle/2, and subtract the other terms in (3.18). Applying (3.26) gives

Completion of the square proves (3.23).

Averaging. For the averaging, set

Independence and sign symmetry kill every term with an odd number of occurrences of either random field. Exchangeability identifies the two pure and the two unmixed terms. Expansion in the four slots therefore gives

Both \int_0^1((1-s)^2+s^2)\dd s and \int_0^14s(1-s)\dd s equal 2/3. The contribution of the P term to \mathbb E K_{AB} is consequently (T+S-H)/3. In the radial case C_s=sA, D=A, Q_s=-s\nabla_AA, so the square vanishes and K_{0A}=P(A,A;A,A)/6. Finally,

These facts prove (3.24).

The curvature contraction. To identify the curvature convention explicitly, write

Separate symmetric polarization of (3.22) gives

whereas

Exchangeability gives \mathbb E|U|^2=\mathbb E|V|^2 and

Hessian symmetry for the gradient A gives \langle A,U\rangle=\langle B,a\rangle; swapping the random choices then yields \mathbb E\int L_B\langle A,U\rangle =\mathbb E\int L_A\langle A,b\rangle. Thus

The first two terms of the specified integrated curvature formula, with (X,Y,Z,W)=(A,B,B,A), combine to

Its next two terms combine to

Finally [A,B]=V-U, and Hessian symmetry for B gives

The sum is exactly the integrand in (3.28). This proves (3.25), including the sign of the bracket term.□

Proof. First keep all endpoint corrections S_A fixed. In twice the expected radial expansion from Lemma 3.5, minus the expected pair lower bound, every term displayed before order four cancels. Indeed independence and \mathbb E u_A=0 give the exact identity

This identity uses no further expansion of the endpoint densities. The quadratic terms cancel by (3.2). The radial cubic term is a constant multiple of \int f\langle A,\nabla_AA\rangle, which is odd under sign reversal. The pair cubic term is a constant multiple of

which is odd under simultaneous sign reversal of both choices. Their expectations therefore vanish. Together with Lemma 3.3, this proves

Now approximate -\nabla_AA in L^2 by admissible fields S_A^{(j)}. Such an approximation can be chosen explicitly: |A|^2/2 is bounded, compactly supported, and Sobolev; heat regularize it, multiply by a fixed test cutoff equal to one on a neighborhood of its support, and take the negative gradient. Strong continuity of the heat flow in W^{1,2} gives

Each approximant is a bounded compact test gradient with bounded divergence. Its bounds may depend on j. For every fixed j, the limit h\downarrow0 leading to (3.30) has already been taken, so no uniform flow bounds in j are required.

The dependence of K_{AB}(S) on an endpoint correction is a pairing with one of the fixed L^2 fields fQ_0,fQ_1. It is therefore continuous under this approximation. The radial defects tend to zero. Taking j\to\infty in (3.30) gives

Lemma 3.6 is exactly (3.29).□

Proof. Use the inf-convolutions

Minimizers exist by properness. If L=\operatorname{Lip}(f), then

In particular the supports lie in the fixed compact support of f. Every minimizer y satisfies d(x,y)\leq\varepsilon L: compare it with a point a short distance along a minimizing segment from y to x, divide the resulting optimality inequality by that distance, and let the distance tend to zero. The same bound in the difference-quotient estimate for the inf-convolution gives local slope at most L; since M is a length space, \operatorname{Lip}(f_\varepsilon)\leq L.

For sufficiently small fixed \varepsilon, all these minimizing distances are below a fixed model-comparison scale. Triangle comparison makes the functions x\mapsto d(x,y)^2 uniformly semiconcave when these distances remain below that scale, including at x=y; compare [28, Section 1.2]. For an explicit normalization, choose a>0 with -a^2\le\kappa, weaken the curvature bound to -a^2, and put h(r)=(\cosh(ar)-1)/a^2. Model comparison gives (h\circ d_y)''\le1+a^2h\circ d_y along unit-speed geodesics. Write r^2=\Psi(h(r)). The function \Psi is increasing and concave, is smooth at zero, and satisfies 0\le\Psi'\le2. Thus on d_y\le R the squared distance has semiconcavity constant 2\cosh(aR), uniformly in the center. Locally near each x, the infimum defining f_\varepsilon may be restricted to a common small neighborhood of x, by the minimizer bound. It is thus an infimum of functions with a common semiconcavity constant there. Infima preserve that semiconcavity inequality. The constant depends on \varepsilon,\kappa,L, but not on the base point. Local semiconcavity with this constant yields the same inequality along each geodesic by subdivision. This proves the required semiconcavity of each f_\varepsilon.

The common support and Lipschitz bound give boundedness in W^{1,2}. Uniform convergence identifies every weak Sobolev subsequential limit with f; hence any sequence \varepsilon_j\downarrow0 has the asserted weak convergence. In (3.29), the two terms without curvature are integrals against fixed locally integrable functions. The curvature term converges by Lemma 2.3. Passing to the limit proves the last assertion.□

Proof. The tangent module is generated by gradients of Sobolev functions. Thus finite sums of such gradients with bounded measurable scalar coefficients are dense in L^2(TM). Approximating the coefficients by finite-valued functions and refining the resulting partitions shows that fields of the form

are dense as well: on each piece a constant linear combination of gradients is the gradient of the corresponding linear combination of potentials. This construction may be made simultaneously for the finite list V_r. Since f is bounded, it provides a finite measurable partition and choices q_{r,\ell} for which

Put K_f=\operatorname{supp}f. On this compact set the measure with density

is finite and regular. Choose disjoint compact subsets K_\ell\subset E_\ell\cap K_f whose complement has arbitrarily small measure for this density. There are nonnegative Lipschitz functions \alpha_\ell summing to one, with \alpha_\ell=1 on K_\ell. For example, take Lipschitz bumps equal to one on the nonempty K_\ell, supported in disjoint neighborhoods, and assign their remaining complement to the first function. On \bigcup_\ell K_\ell, these functions agree with the corresponding partition indicators. On the complement, the change in the error integral is bounded by the sum appearing in the displayed density. Taking the omitted measure sufficiently small proves (3.31).□

Proof. First retain an arbitrary finite sign-symmetric law as above. Apply Lemma 3.9 to the finite list

For each \ell, apply (3.29), as extended in Lemma 3.8, with weight f\alpha_\ell and potentials q_{AB,\ell}. Sum over \ell. The curvature functional and its model term are linear in the weight, and \sum_\ell f\alpha_\ell=f. Thus their sums are exactly their evaluations at f, including the terms containing derivatives of the weight. The error is at most \eta/2, after decreasing the tolerance in (3.31) if needed. Letting \eta\downarrow0 gives

In particular there are no partition-derivative errors and no uniform derivative bound on the partitions is needed.

Choose the uniform law on the four potentials yielding the fields U,-U,V,-V. In (3.33), pairs from the same signed field have zero sectional and wedge expressions. The mixed pairs have total probability 1/2. Multilinearity and the curvature symmetries identify all of them with the corresponding expression for (U,V). Both sides of (3.33) are therefore one half of the respective sides of (3.32). This proves the proposition.□

Proof. For constant coefficient vectors a,b\in\mathbb R^N, put E(a)=\sum_i a_iE_i. This is again in \mathcal G. Proposition 3.10 says that

is nonnegative on nonnegative compact Lipschitz functions. It is therefore a positive Radon measure. Indeed, for weights supported in a fixed compact set, choose a nonnegative Lipschitz cutoff dominating one there. Positivity bounds the functional by its value on that cutoff times the uniform norm of the weight. The functional extends to continuous compactly supported functions and the Radon representation theorem applies.

Define functionals on compact Lipschitz weights by

For \alpha,\beta\in\mathbb N_0^N with |\alpha|=|\beta|=2, let

Here e_i are the coordinate vectors, and the first sum is over ordered quadruples, retaining all monomial multiplicities. Polynomial interpolation at finitely many constant vectors expresses every grouped coefficient C_{\alpha\beta} as a finite real linear combination of the positive Radon measures T_{a,b}. Thus each C_{\alpha\beta} is a signed Radon measure; no measure representation of an individual D_{ijkl} is needed.

On a fixed relatively compact region containing the support of the weight, put

For every rational pair a,b, positivity of T_{a,b} gives P_x(a,b)\geq0 for \lambda-almost every x. Countability and polynomial continuity give a common conull set on which this holds for every real pair.

For variable test coefficients, first apply tensoriality of R over the test algebra. All scalar weights in the following identity are compact test functions:

The last line uses only the continuous representatives of scalar test functions on the signed measures. No tangent field is evaluated on a singular measure support. This proves the claim, including when some C_{\alpha\beta} are singular with respect to m.□

Proof. Choose a fixed compact neighborhood E' of E and apply Lemma 2.1(iii) on E'. It gives the uniform gradient bound and all the asserted scalar convergences on this fixed neighborhood. Strong L^2 convergence of the gradients gives almost-everywhere convergence after passage to a subsequence.

For X=\sum_i a_i\nabla b_i, keep the original coefficients and approximate its finitely many potentials as just described. The product rules give

The coefficients and their gradients are bounded, so both expressions converge strongly in L^2 on the neighborhood. The vector bounds and convergence follow from the corresponding properties of the gradients. Use one common subsequence for the finite collection.□

Proof of Theorem 3.1. It remains to extend Proposition 3.10; the RCD and support conclusions were noted at the beginning of the section. Fix a nonnegative compactly supported test weight f and arbitrary X,Y\in\TestV(M). Lemma 3.12 gives approximating fields X_j,Y_j near \operatorname{supp}f. Lemma 3.11 proves (3.1) for every pair X_j,Y_j.

The field-continuity assertion of Lemma 2.3 passes the curvature evaluation to the limit on the fixed compact support of f. The wedge term converges by dominated convergence on that support. This proves (3.1) for arbitrary test fields and compact test weights.

Finally let f\in\Test(M) be nonnegative without a support restriction. Choose 0\leq\chi_R\leq1 in the stronger test cutoff class, equal to one on B_R(o), with compact support and uniformly bounded gradient. Such an exhaustion exists for every fixed K,n [19, Appendix A, Lemma A.2]. The product f\chi_R is a compact test function, so the desired inequality holds for this weight.

All the original tensor integrands are globally integrable. Test fields are bounded and in L^2, and their derivatives and divergences are in L^2. Terms with two derivatives are thus L^1; a typical term with df is bounded by a constant times |W||\nabla Z|, again in L^1. The new terms containing d\chi_R obey the same bound, with an additional indicator of M\setminus B_R(o), and hence their integrals tend to zero. The model term is integrable, since

Dominated convergence proves (3.1) for the original global weight. This completes the proof.□

Proof. The first inequality is the L^1 gradient estimate [31, Corollary 3.5]; the second is the reverse Poincaré inequality. The Bakry–Ledoux estimate [12, Proposition 4.9] gives the third: its coefficient of |\Delta P_sh|^2, before rearrangement, is (1-e^{-2Ks})/(nK), with value 2s/n at K=0. Apply (4.6) at time s/2 to P_{s/2}h, then apply (4.5) inside the remaining heat operator, to obtain (4.7).

These inequalities give the L^\infty-to-Lipschitz and L^\infty-to-L^\infty Laplacian bounds at positive time. Semigroup splitting, together with commutation in L^2, therefore gives, for each integer j\ge0,

The time dependence is differentiable in the corresponding L^2 generator domains on intervals away from zero.

Here is a local extension argument that does not assume m(M)<\infty. Choose 0\le\chi_R\le1, equal to one on B_R(o), compactly supported, and set h_R=\chi_Rh. For R'\ge R, the difference d_R=h_R-h_{R'} vanishes on B_R(o) and satisfies |d_R|^2\le\|h\|_\infty^2\mathbf1_{M\setminus B_R(o)}. The Gaussian kernel bound shows that P_s(d_R^2) tends to zero uniformly on compact spatial sets and compact subintervals of s>0. Equations (4.5) and (4.7) then make the gradients and Laplacians Cauchy locally in L^\infty, as well as the functions themselves. Distributional closure identifies their limits with those of P_sh. Repeated splitting gives the same local conclusions for generator powers. The kernel bounds used here are valid for all centers, with constants depending only on K,n,S; see [20, Theorem 1.2].

For integrable nonnegative data, the general-measure Li–Yau inequality is [19, Theorem 3.4]:

Its value at K=0 is understood by continuity. We may weaken the Ricci bound to a strictly negative constant. Rearranging the inequality, using |e^{2Ks/3}-1|\le Cs and the C/s bound for the remaining coefficient on s\le1, gives (4.8). Apply this to h_R and pass to the local limit just established. Positive-time positivity and continuity supply a lower bound on each compact region, so division and the logarithmic chain rule pass to the limit. This proves the assertion also for bounded data that are not in any finite L^p space.□

Proof. The first heat datum is a constant plus a compactly supported W^{1,2}\cap L^\infty function. Constants are preserved by the heat flow, so Lemma 4.2 applies. Since u is bounded, H_t has positive global lower and upper bounds for fixed c. Thus g=f/H_1 belongs to L^1\cap W^{1,2}\cap L^\infty, and the same lemma applies to J. Positivity of the kernel gives J>0; its continuous representative has a positive minimum on each compact space-time region under consideration.

The logarithmic Laplacian formula

and its counterpart for J give local boundedness and W^{1,2} regularity. The squared-gradient term is Sobolev by metric compatibility and the L^2 Hessian estimate. Product and chain rules similarly show that \rho is a local test function. Generator differentiation before composition, followed by these same rules, gives strong local differentiation in the Laplacian graph norm; the Hessian estimate (2.3) then gives the Hessian statement.

If L_u=\Lip(u), then |\nabla e^{-u/(2c)}|\le (L_u/(2c))e^{-u/(2c)}. Use (4.4), applied to the compact perturbation of the constant, to get |\nabla\phi_t|\le e^{-Kct}L_u. Also

Applying (4.4) to g gives the corresponding bound for \nabla\psi. This proves (4.14) and also bounds the gradients of the terminal expressions. Since \rho=HJ, \nabla\rho=\rho w_c/c; differentiating the two factors proves (4.15)–(4.16) with the displayed signs.

Symmetry and the semigroup property, with nonnegative integrands, give

The maximum principle bounds \phi_t between \inf u and \sup u. Finally J_t\to g strongly in W^{1,2}. Heat continuity for bounded Lipschitz data, using the uniform small-time first moment of the kernel, gives uniform convergence of J_t to g. Strong local convergence for the products and logarithms follows from their local positive lower bounds. This proves (4.17).□

Proof. The two-sided Gaussian estimate and Bishop–Gromov give constants a,C,c_0>0, uniform in the base point, for which

The harmless bounded factors involving s\le1 are absorbed in the constants. In addition, for j\ge0,

The lower bound and the logarithmic Lipschitz condition give P_sg(x)\ge c_0e^{-A}g(x). On the annulus j\sqrt{s}\le d(x,y)<(j+1)\sqrt{s}, g(y)^2\le g(x)^2e^{2A(j+1)}. Therefore

This convergent series is uniform in x,s. Dividing the two estimates proves the claim.□

Proof. Bounded compression. Apply (4.8) at heat time ct, using (4.14). It gives

The constants are independent of c. Since v_c=\nabla\phi+w_c, the density equation can also be written

For p\ge2, multiplication by p\rho^{p-1} gives

For fixed c the integrations are justified first with exhausting cutoffs: \rho is bounded and integrable, |\nabla\rho|\le C_c\rho, and \Delta\phi is globally bounded away from the initial time. The cutoff errors vanish by integrability. One may equivalently use the strong interior L^2 heat differentiation before removing the cutoffs. Integrate backward from \rho_1=f, then let p\to\infty. This proves the density bound, with the direction of integration fixed by (4.20).

The terminal lower bound. For the endpoint bound let r=1-t, g=f/H_1, and s=cr. When r\le c, the logarithmic Lipschitz bound for g is at most C/c, and s\le c^2. Lemma 4.4 and (4.6) consequently give

For r\ge c, use the preceding Li–Yau bounds instead. Since b=(\psi-\phi)/2, one can thus take, after increasing constants depending on \delta,

For sufficiently small c, its integral is at most C_\delta(c|\log c|+c). The Sobolev function \Delta b has the same lower bound outside a polar set in its quasi-continuous representative, by the capacitary facts in Section 2.□

Proof. The integrated Bochner identity. Choose a nonnegative space-time cutoff \chi, zero for t\le\delta/2, equal to one on [\delta,1]\times E, and compactly supported in space, such that

For example take fourth powers of a smooth time cutoff and a good spatial test cutoff. We first work on an interval with terminal time T<1. The equations and the definition of \Gamma_2 yield

To check the last term, put q=\Delta b. Twice integrating by parts, using \nabla\rho=\rho w_c/c, gives

This identity is distributional; q is locally Sobolev and bounded, and the required multipliers are admitted by the Bochner measure calculus. Thus () does not assume a pointwise fourth derivative.

Weighted second-order bounds. Average (2.5) and (2.6). For either z=b,\theta,

The two cutoff terms in () are bounded by \eps\int\rho\chi(|\Delta b|^2+|\Delta\theta|^2) plus a constant, using (4.14), (4.19), and Young’s inequality. The possibly negative K-terms have the same harmless bound. At the initial endpoint the primitive is zero. At the terminal endpoint its limit is

which is bounded uniformly in c. This follows from (4.17); no Hessian of \log f is needed. After absorption, let T\uparrow1. Monotone convergence for the squared terms gives the first four bounds in (4.21). In particular these bounds also hold uniformly for all shorter terminal intervals.

The fourth-order bound. Choose a spatial cutoff \eta equal to one on E, with |\nabla\eta|\le C\sqrt\eta. The divergence of \rho\eta|w_c|^2w_c gives

All terms are integrable on interior intervals by local test calculus. Integrate in time and then exhaust the terminal interval. If A_c=\int_\delta^1\int\rho\eta|w_c|^4, Cauchy–Schwarz and the bounds just proved show A_c/c\le C\sqrt{A_c}. For the cutoff term use also the uniform bound on w_c and the mass bound. Hence A_c\le C^2c^2, proving the last estimate.

Vanishing weighted products. It follows that \int_\delta^1\int_E\rho^c|w_c|^2\to0. For (4.22), truncate the fixed function T: on \{|T|\le A\} use this convergence, and on its complement use the uniform bounds on \rho^c,w_c. The latter contribution is bounded by C\int_{\{|T|>A\}}|T|^2, which tends to zero as A\to\infty.□

Proof. All assertions are local, so first insert the cutoffs just described. Since \nabla_a a=H_za=\nabla(|a|^2/2), the first two terms of the defining expression for R(Y,a,a,Y)(h) sum to H_{|a|^2/2}(Y,Y)(h). Its other terms sum to

because [Y,a]=H_zY-\nabla_aY. This is exactly (4.25). The calculation uses Hessian symmetry and first-order products only; it does not require a third covariant derivative of z.

For fixed c, on a compact interior time interval the pairing on the left of () is absolutely continuous. This follows from local graph-norm differentiation of \theta, the Hessian estimate, and local boundedness of \partial_t\rho. Differentiate that pairing, use \partial_t\rho=-\divg(\rho v_c), and interpret its contraction against H_\theta through (4.24). Thus no continuity equation is being tested against an unproved Sobolev representative of H_\theta(Y,Y). Substitute (4.16) and apply (4.25) to z=\theta. The \nabla_{v_c}H_\theta terms cancel, leaving ().□

Proof. Exact cancellations. We work first at fixed c and at an interior time, suppressing these parameters. By (4.25) with z=b, the error is the sum of

The first term is nonnegative. The curvature bound gives R(Y,w,w,Y)(\rho)\ge \kappa\int\rho|Y\wedge w|^2\dd m, whose possible negative contribution tends to zero after time integration by (4.22).

We examine I. Formula (4.24) and \nabla\rho=\rho w/c give

Put D=\divg Y, A=\nabla_YY, and \alpha=Yb=\langle Y,w\rangle. Twice using the definition of the distributional Hessian gives

The last divergence is a measure, not necessarily a function. Its product with \rho c\Delta b is understood using quasi-continuous representatives.

To justify () for the actual test class, observe that the function \divg(\rho Y)=\rho D+\rho\alpha/c belongs locally to W^{1,2}. The product rule for its product with the bounded field Y, proved first after truncation, gives an L^1 divergence. The multiplier \Delta b is locally bounded Sobolev, so the truncations pass by L^2 pairing for the gradient term and dominated convergence for the divergence term. Also A=\nabla(|Y|^2/2), whose divergence is the Bochner Laplacian measure; multiplication by the Lipschitz function \rho obeys its usual weak product rule. Expanding Y\alpha=H_b(Y,Y)+\langle w,A\rangle gives the second line of ().

The terms \rho\Delta b H_b(Y,Y) in (4.28) and () cancel. There is a second exact cancellation:

Indeed integration by parts in the first term on the left produces -\rho\alpha^2|w|^2/c-2\rho\alpha H_b(w,Y) -2\rho\alpha\langle w,\nabla_wY\rangle. Writing H_b(Y,Y)=Y\alpha-\langle w,A\rangle and integrating the negative Hessian term gives the opposites of the first two expressions, which cancel them, with the remaining terms displayed on the right of (4.30).

The remaining Sobolev terms. After division by c, each term on that right side is bounded in absolute value, on the whole time interval, by a constant times

where E contains the common support and T_0 is a fixed L^2 coefficient formed from \nabla Y,D and bounded trial fields. This tends to zero by Lemma 4.6. The remaining terms involving H_b or \Delta b, one factor w, and derivatives of Y are bounded the same way, using their weighted L^2 bound in place of the first factor.

The Bochner measure term. It remains to control the final parenthesis in (). Here D is only L^2, so D^2 need not be an L^2 coefficient. We retain it in a nonnegative measure, to which the lower bound for c\Delta b applies, and leave an L^2 remainder. Since Y=\nabla g_t, the required decomposition is

Here \ell_t has a uniformly bounded local L^2 norm. The measures \mathcal M_t have uniformly bounded mass on E: polarize \Gamma_2 on the fixed finite list of test potentials and use boundedness of the smooth temporal coefficients. Equivalently, test their defining formula with a fixed outer cutoff. Thus no L^4 bound on D is used.

Time measurability and terminal uniformity. To justify time integration of the singular part, fix a compact neighborhood E_* of the trial support and define the finite signed coefficient measures and their common dominating measure by

Every coefficient measure neglects polar sets. The identity \mathcal M_t=\sum_{i,j}a_i(t)a_j(t)\mathcal M_{ij} shows that \mathcal M_t|_{E_*} has a jointly measurable, uniformly bounded \Lambda-density.

Choose a compact test cutoff \chi=1 near E_*. At fixed c, on each compact interior time interval, \chi\rho_t c\Delta b_t is strongly continuous in W^{1,2} and has a common L^\infty bound. Indeed, for either positive heat factor U, the spatial formula

expresses the derivative using bounded Lipschitz heat powers, local positive lower bounds, and the strongly locally L^2-continuous Hessian H_U. The latter continuity follows from the Laplacian graph estimate; continuity of the other factors follows by integrating the next generator powers. The fixed cutoff and \rho_t preserve these Sobolev continuity and boundedness properties.

Strong Sobolev convergence with a common bound has quasi-everywhere convergent subsequences. Since \Lambda is finite and neglects polar sets, dominated convergence and the subsequence criterion give continuity of these quasi-continuous multipliers in L^1(\Lambda). Choose a jointly measurable representative of this curve. At almost every time it agrees \Lambda-almost everywhere with the quasi-continuous multiplier. The lower bound for c\Delta b_t therefore holds \Lambda-almost everywhere at each admissible time, and Fubini applies without an uncountable union of polar exceptional sets. All the bounds in this paragraph are only fixed-c, interior-time bounds.

The quasi-continuous lower bound in (4.19), and the density bound there, give

Bochner measures neglect polar sets, so an almost-everywhere bound on the Sobolev multiplier has been used here only after passing to its quasi-continuous representative. In particular, for every T<1,

This lower error bound tends to zero uniformly in T. The absolutely continuous remainder is bounded by

The Sobolev remainders were estimated in absolute value on (\delta,1), and the remaining measure contribution was bounded below using \|e_c\|_{L^1(\delta,1)}. After discarding the nonnegative terms, these bounds hold with the same error for every T<1, proving (4.27).□

Proof. Integrate () from \delta to T<1. The initial pairing vanishes. Apply (B) to its curvature term and use

Lemma 4.8 bounds the last term uniformly in T. The resulting right-hand side is integrable up to one, by boundedness of \rho,v_c and the fixed L^2 derivatives of Y.

For the terminal left-hand side, use the first-order expression (2.4). The convergences (4.17) imply strong local L^2 convergence of \divg(\rho_tY_t) to \divg(fY_1), and of the gradients of \theta_t to that of \phi_1^c+c\log f. The bounded density and field factors converge uniformly on the compact support. Both products in (2.4) therefore converge in L^1. Let T\uparrow1, with c still fixed. This proves (4.32). The remaining limits will be taken successively: first c\downarrow0, and then removal of the positive terminal perturbation \lambda\bar f by \lambda\downarrow0.□

Proof. Use the two-sided Gaussian bounds with exponents 4-\eta and 4+\eta, for any fixed 0<\eta<1, and heat time ct. The kernel is symmetric, so the volume factor can be taken to be m(B_{\sqrt{ct}}(x)) at the evaluation point. These are the bounds of [20, Theorem 1.2], after weakening the Ricci bound to a negative one if needed.

Here are the two Laplace estimates with their uniformity specified. Fix t\in[\alpha,1] and x in a compact set. In the upper kernel bound reserve a small fixed fraction \tau>0 of its Gaussian exponent. Taking the infimum of the other part of the exponent gives

Indeed the integral of the reserved Gaussian, divided by m(B_{\sqrt{ct}}(x)), is uniformly bounded by the annular volume-comparison estimate used in Lemma 4.4. For the lower kernel bound, choose a minimizer for the corresponding modified quadratic cost and integrate over a small ball of fixed radius r>0 around it. Boundedness of u keeps all these minimizers in a common compact set. The Lipschitz bound for u, and the bounded distance of the minimizers from x, change the exponent by at most C_\alpha r. The logarithm of the volume prefactor, multiplied by c, tends to zero uniformly: on the compact region, Bishop–Gromov and full support bound small-ball volumes below by a fixed multiple of the n-th power of the radius, and above by a fixed finite constant. Thus

uniformly on the compact parameter set. Send first c\downarrow0, then r,\eta,\tau\downarrow0. The quadratic infima vary uniformly with their time parameter on compact positive-time intervals, since their minimizers have uniformly bounded distance. The two estimates give (4.33). Finally the uniform spatial Lipschitz bound gives weak local Sobolev compactness; closure of the differential identifies its limit with \nabla\Phi_t.□

Proof. Let L be the velocity bound in (4.14) and r(x)=d(o,x). For a bounded nondecreasing Lipschitz function \chi that is constant at large arguments, the continuity equation, first with compact spatial tests and then after subtracting that constant, gives for s\le t\le1

Compact Lipschitz tests are admitted by their strong Sobolev test approximations. At time one, local convergence of the density and preservation of its total mass justify the limit for these tests. Hence

Approximate r^2 and then r^2\mathbf1_{\{r>R\}} monotonically by such functions. The first choice proves finiteness of the moments, and the second gives

The same bound also gives tightness.

The explicit heat factors give continuity against compact tests on the interior and at the terminal endpoint. Tightness gives narrow continuity, and (4.35) upgrades it to W_2 continuity. The weak continuity equation extends to every h\in W^{1,2}, with functional

To see this extension, approximate h in W^{1,2} by compact tests; the density bound makes the errors uniform in time on [\delta,1].

For the exact domain in the cited continuity-equation criterion, write S^2(M) for the Sobolev class with square-integrable weak gradient, without requiring the function itself to belong to L^2. The same formula defines \mathcal L_t^c on all of S^2(M), with the displayed dual-seminorm bound. The integrated continuity equation holds for every h\in L^1(m)\cap S^2(M): set h_M=\max\{-M,\min\{h,M\}\}. Then |h_M|^2\le M|h|, so h_M\in W^{1,2}, and

Locality makes the choice on the threshold level sets irrelevant. Bounded compression gives

while, on any fixed positive-time interval I,

Thus the integrated identity passes for every pair of times, with finite scalar integrals and an absolutely continuous representative. The identical argument applies to the limiting gradient functional with \nu,V below, since it has the same density and velocity bounds. The continuity-equation criterion now gives |\dot\mu_t^c|\le L almost everywhere. Finally properness, the uniform moment tails, and Arzelà–Ascoli in \mathcal P_2(M) give the claimed subsequence. The bound on the limiting densities follows by testing their narrow limit against compact continuous nonnegative functions. A diagonal choice handles intervals with arbitrarily small positive left endpoint.□

Proof. On [\alpha,\beta] the fluxes j_c=\rho^cv_c are bounded in the fixed Hilbert space L^2(\dd t\dd m;TM), since \int|j_c|^2\le C_\alpha\int\rho^c|v_c|^2. Extract a weak flux limit j, and a weak-* density limit \rho^c\rightharpoonup^*\nu in L^\infty(\dd t\dd m). The latter agrees with the curve limit. Testing the convex inequality |j_c|\le L\rho^c against bounded fields of bounded support gives |j|\le L\nu. Define V=j/\nu on \{\nu>0\} and zero elsewhere. The continuity equation passes to the limit. Its kinetic lower semicontinuity follows, for example, from

where the supremum is over bounded L^2 fields with bounded support.

For the first equality in (), write a=\nabla\phi and use v_c=a+w_c. The equations give

because c\int\rho\Delta\phi=-\int\rho\langle w_c,a\rangle. One first uses spatial cutoffs. For fixed c, their errors vanish by mass integrability, boundedness of \phi,a, and |\nabla\rho|\le C_c\rho. The identity extends to \beta=1 using the terminal traces and bounded gradients; no Laplacian bound at exactly time one is needed. The difference between the actions of a and v_c tends to zero on bounded spatial regions by Lemma 4.6. On their complement it is bounded by a constant times the density mass, uniformly small by (4.35). This proves the second equality.

Put h=\beta-\alpha and

The bounded potentials, local uniform convergence, and tightness identify this with the limit of the left side of (). The Hopf–Lax semigroup property says \Phi_\beta(y)-\Phi_\alpha(x)\le d^2(x,y)/(2h). The continuity-equation criterion and action lower semicontinuity therefore give the sandwich

Thus all inequalities are equalities. Equality in the metric action bound proves the geodesic assertion and equality of the velocity norms.

To turn norm convergence into (4.37), expand the square. For its cross term, test the weak flux convergence against V\mathbf1_{B_R(o)}; for its last term, test the density convergence against |V|^2\mathbf1_{B_R(o)}. These are fixed admissible tests in the appropriate spaces. Boundedness of the velocities and tightness let R\to\infty. The resulting limit is zero by equality of actions in ().

Finally, for any A_0>0,

First let c\downarrow0, then A_0\to\infty. This proves (4.38).□

Proof. Initially normalize f\in\mathcal F to mass one. Equality () on every positive-time interval makes \mu_t a consistent constant-speed Wasserstein geodesic. Its speed is bounded, so completeness of \mathcal P_2(M) extends it to time zero. A dynamical optimal plan represents this geodesic; one may use metric superposition [17, Proposition 2.1] and equality of endpoint cost and path action. Since \|Q_tu-u\|_\infty\le t\Lip(u)^2/2, the dual equalities in () pass to time zero and give

The nonnegative pointwise duality gap therefore vanishes on almost every path, proving (4.2) for this plan. Its individual path lengths are bounded by \sqrt{2\operatorname{osc}(u)}. The positive-time compression is the limiting bound \nu_t\le C_\delta.

We specify the identification with V. On a positive-time interval the plan is a test plan. Its pullback velocity, defined by differentiation of Sobolev functions, has norm equal to the metric path speed [14, Theorem/Definition 2.33]. Let W be its conditional average given (t,\gamma_t). It solves the same continuity equation and its action is at most the path action, by conditional Cauchy–Schwarz. The kinetic lower bound makes this an equality. Both V and W solve the same equation; apply the same lower bound to (V+W)/2 and use the parallelogram identity. Equality of their actions with the minimum forces V=W. Equality in conditional Cauchy–Schwarz then identifies the pullback velocity itself with V_t(\gamma_t) almost everywhere. Separability of the tangent module, available on this RCD space, permits the conditional-average construction by a countable family of module generators.

The prescription is independent of the positive datum. Take two normalized calibrated plans, reverse their time orientation, and form their equally weighted mixture. The mixture is still calibrated, and hence is optimal from its absolutely continuous initial measure, which is the mixture of the two terminal densities, to its other endpoint measure. The latter has finite second moment and may be singular. Uniqueness of dynamical optimal transport from an absolutely continuous measure [18, Theorem 1.1] forces this reversed mixture to be induced by a map into whole geodesics. Disintegration consequently forces both component plans to choose the same path wherever their terminal densities overlap. Positive members of \mathcal F are equivalent to m, so this agreement holds m-almost everywhere.

Fix the reference datum proportional to \bar f and choose its resulting Borel map S. The argument just given shows that every heat limit for every positive datum is S_\#(fm), after restoring mass. The prescribed-potential uniqueness theorem [18, Theorem 1.3] applied to the c-concave potential Q_1u, c=d^2/2, also gives uniqueness of the calibrated path for almost every terminal point.

For a compact datum f_0\ge0, the ratio f_0/\bar f is bounded. Thus S_\#(f_0m) is boundedly dominated by the reference plan. This proves its compression assertion and transfers its path-velocity representative. In particular one can choose this representative using the reference plan once and for all. Restriction to a bounded set of terminal points preserves all these statements and, by the speed bound, gives bounded spatial support on positive-time intervals.□

Proof of Theorem 4.1. First take f\in\mathcal F. On the common compact support of the trial, Lemma 4.10 and the uniform gradient bounds imply

Indeed (2.4) pairs the gradient of its potential with fixed L^2 coefficients; weak local Sobolev convergence suffices. The additional gradient c\nabla\log f converges uniformly to zero.

For the right side of (4.32), use (4.38) with the fixed coefficient \nabla Y. It replaces \nabla_{v_c}Y by \nabla_VY strongly in the weighted L^2 norm. The remaining fixed squared expression is in L^1(\dd t\dd m) and has compact support, so weak-* convergence of the densities passes its integral to the limit. The curvature term passes in the same way using boundedness of Y and the velocities. Lemma 4.13 identifies the limiting expression with the path integral in (4.3). Let c\downarrow0 in Proposition 4.9.

Finally, for the desired compact datum f_0, use f_\lambda=f_0+\lambda\bar f. The common prescription gives \pi_{f_\lambda}=\pi_{f_0}+\lambda\pi_{\bar f}. Both sides of the already established inequality are linear in the terminal data. All terms involving \bar f are finite, by compact support of the trial, bounded speed, and positive-time compression. Let \lambda\downarrow0. This proves (4.3), including the zero datum.□

Proof. Bounds along dyadic radii imply bounds at every radius by enlarging to a dyadic radius at most twice as large. Fix \varepsilon>0, to be chosen sufficiently small in terms of s, and put

The sets E_j cover E. Fix x\in E_j and R<1/j. A content cover of E_j\cap B(x,R) can be recentered in E_j: discard covering sets not meeting this set and replace a covering set of positive diameter d_i by a ball of radius 2d_i centered at a point of its intersection with E_j. Replace singleton covering sets by balls with arbitrarily small positive total s-cost. Choosing the original cover within an arbitrarily small error of the content gives child balls B(x_i,r_i) with

provided \varepsilon was chosen small enough. In particular r_i\leq\theta^{1/s}R<1/j.

Repeat this procedure on E_j\cap B(x_i,r_i), which may enlarge the part of the parent set being covered but still covers it. At generation k the total diameter cost is at most 2^s\theta^kR^s, and every diameter is at most 2\theta^{k/s}R. Both bounds tend to zero. Thus \mathcal H^s(E_j\cap B(x,R))=0. Separability supplies countably many such root balls covering E_j. Taking the countable union over j proves the claim. No finiteness assumption on \mathcal H^s(E) is used.□

Proof. The volume lower bound makes the average of P on B(y,r) within 1/4 of I, simultaneously for all y\in B(x,r), if \varepsilon_0 is small enough. Let \lambda be the enlargement factor in the Poincaré inequality, and choose L>2+5\lambda. For t_j=2^{-j}r, the difference between the averages on B(y,t_j) and B(y,t_{j+1}) is at most

If, for every j\geq0,

the sum of these differences is bounded by

Because s>n-2, this is less than 1/2 for an appropriate c_0>0. The averages converge to the precise value at y, so this contradicts |P(y)-I|\geq1. Consequently, each y\in E has a radius t_y\leq r for which

The 5r covering lemma selects countably many disjoint balls B(y_i,\lambda t_i) whose fivefold enlargements cover E. They lie in B(x,Lr). Summing (5.3) over these disjoint balls bounds \sum_i t_i^s by the right-hand side of (5.2), up to a fixed factor. Their fivefold enlargements give the required content cover.□

Proof of Proposition 5.1. Selection at all dyadic scales. Put r_k=2^{-k}, and choose a countable r_k-net N_k in M. For each integer A\geq4, each y\in N_k, and each integer q\geq1, select one harmonic n-tuple on B(y,2Ar_k) with splitting error at most 1/(n^2q), whenever such a tuple exists. The splitting-map estimates give a dimensional gradient bound and

for its components u^a and their Gram matrix P. The existence input is the harmonic splitting-map theorem near a Euclidean rescaling; see [8, Proposition 1.4]. To match that theorem’s product formulation, use the auxiliary synthetic dimension N=n+1, Euclidean factor dimension n, and a point as the \RCD(0,1) residual factor. Its radius is 2Ar_k/5, so its output ball is B(y,2Ar_k). The rescaled curvature bound tends to zero. This auxiliary parameter does not change the actual dimension n, the measure \mathcal H^n, or the volume estimates used here.

Subtract the values u^a(y) and multiply by a good test cutoff supported in B(y,2Ar_k) and equal to one on B(y,Ar_k). These are compact test functions. Indeed, on the support of the cutoff \eta, the functions and their gradients are bounded and their Hessians are locally in L^2, and

The right-hand side is in W^{1,2} by the product rule and the local Hessian bounds. The construction therefore changes none of the Gram entries on the inner ball. The collection of all selected components is countable. Adjoin the prescribed countable family.

Fix a regular point x, a bounded region containing it, and an integer A>L+1, with L as in Lemma 5.3. Choose y_k\in N_k with d(x,y_k)\leq r_k. The rescalings at y_k converge to Euclidean space at each fixed enlargement of r_k. To check this, take any subsequence of rescalings at x. Their limit is Euclidean by regularity. The moving basepoints are at bounded rescaled distance and hence have a further convergent subsequence; the resulting Euclidean pointed space is isometric to the one pointed at the origin. This proves the assertion for the original sequence as well. The same reasoning applies to normalized measures.

It follows that, for each fixed q, the selected harmonic tuple with threshold q is available at (y_k,2Ar_k) for every sufficiently large k. Choose q_k\leq k maximal among the available thresholds, ignoring the finitely many initial scales where there is no choice. Then q_k\to\infty. On B(x,Lr_k), the resulting Gram matrix P_k agrees with that of the harmonic tuple. Metric compatibility, bounded gradients, and (2.7) imply

Thus the conclusion holds at every sufficiently small dyadic scale, rather than only along a sparse subsequence.

Common precise representatives. Every pairwise inner product of the countably many gradients is a bounded Sobolev function. On the present complete, locally doubling Poincaré space, such functions have L^2-Lebesgue representatives, given by limits of ball averages outside a set of 2-capacity zero, and these are their quasi-continuous representatives; see [6, Propositions 4.8–4.9]. The equal-norm identification of Newtonian and RCD weak-gradient Sobolev spaces follows from [3, Theorem 7.4 and Section 8.2]; see also [15]. Completeness makes M its own completion, so the path-openness hypothesis of Proposition 4.9 and the measurable-inclusion qualification in the appended correction are automatic here. To identify the polar sets, let C_W be the outer Sobolev capacity and C_N the Newtonian capacity, using the same norm normalization. On an open set U, a Newtonian competitor is a Sobolev competitor of the same norm. Conversely, truncate a Sobolev competitor to [0,1] and take its equal-norm Newtonian representative. This representative equals one at every one of its L^2-Lebesgue points in U. The exception is C_N-polar, so resetting it to one there preserves its Newtonian class and norm. Thus the two capacities agree on open sets. Their outer regularity identifies their polar sets, and uniqueness identifies the resulting quasi-continuous representatives. Remove one common polar set for this countable collection. The rational quadratic forms of the Gram matrices are nonnegative almost everywhere, and all their (n+1)-minors vanish almost everywhere. On the bounded range of any finite list of Gram entries, each such polynomial is Lipschitz. Their L^2-mean convergence at a common Lebesgue point therefore implies that the polynomial averages converge to the polynomial of the precise values. This passes the almost-everywhere positivity and minor identities to these values. The polynomials are themselves bounded Sobolev functions. Removing a further countable union of polar sets makes these identities hold simultaneously in the precise representatives. This proves the positivity and rank assertions outside a single polar set N, which may be taken Borel.

Size of the rank defect. Let S be the set of regular points outside N where no n-tuple is positive definite. The representatives and \mathcal R_n may be taken Borel, so S is Borel. Each chosen n-tuple is singular at every point of S, and hence |P_k-I|\geq1 there. Apply Lemma 5.3 and (5.4). For every regular x and every s>n-2 this gives

The constants are uniform on each bounded region. Lemma 5.2, followed by a bounded exhaustion, gives \mathcal H^s(S)=0. In particular we take s=n-1. When n=2 this is s=1; the telescoping series in Lemma 5.3 has ratio 2^{-1/2}, so no borderline exponent is used.□

Proof. Put T=b-a and M_\pi=\pi(C([a,b],M)). Sobolev differentiation along bounded-compression plans and the elementary one-dimensional bound on the supremum of an absolutely continuous function give

where on the left we initially use the continuous path representative. The chain rule and its representative formulation are part of the RCD Sobolev calculus; see [15, 31].

If N is polar, choose open neighborhoods U_j\supset N and u_j\in W^{1,2}(M) with u_j\geq1 almost everywhere on U_j and \|u_j\|_{W^{1,2}}\to0. For almost every path, membership in the exceptional null subset of U_j occupies zero time, by bounded compression and Fubini. A path hitting U_j has a nontrivial one-sided time interval in U_j. Its continuous representative of u_j therefore has supremum at least one. Estimate (5.5) shows that the measure of paths hitting N is zero.

For the Hausdorff assertion set L_*=\max\{1,L\}. A path hitting B(z,r) spends at least r/L_* units of time in B(z,2r) whenever r<L_*T. This remains true for a hit at an endpoint, using a one-sided interval. Consequently

The last constant is uniform on the bounded region under consideration. Covering an \mathcal H^{n-1}-null set by balls with arbitrarily small total (n-1)-cost proves its avoidance.

Finally, approximate each of the specified Sobolev functions sufficiently rapidly in W^{1,2} by Lipschitz functions. A subsequence converges to its quasi-continuous representative outside a polar set. Estimate (5.5), applied to successive differences, makes the approximations converge uniformly along almost every path. Those paths avoid the polar set just obtained; their uniform limit is therefore the quasi-continuous representative at every time. It also equals the Sobolev path representative, since they agree for almost every time. Take the countable intersection of the resulting conull sets.□

Proof. Almost every path is regular at the endpoints and at every time in a fixed countable dense subset of (a,b). Consider one such nonconstant path and fix an interior time t. Take any sequence of rescalings converging to a tangent Y at \gamma_t. At a nearby regular time t_j, the same rescalings converge to Euclidean n-space. Deng’s estimate, valid along every geodesic, implies for each fixed radius R that

The constants can be kept fixed while t_j stays in a compact interior interval containing t; see [11, Theorems 1.1–1.2]. Letting regular t_j tend to t proves that every pointed ball in Y is Euclidean. The tangent sequence was arbitrary, so \gamma_t\in\mathcal R_n. In the noncollapsed setting the normalized tangent measures are the corresponding Euclidean measures. Constant paths are already covered by regularity at one fixed time. Endpoint regularity came directly from bounded compression, not from the interior estimate.□

Proof. Fibers and connection. Lemmas 5.4 and 5.5 ensure that, along almost every entire path, the common Gram representatives have rank n at every time. Sobolev differentiation of their entries gives

All terms are in L^2 in time along almost every path, since the fields are bounded, their covariant derivatives are in L^2(m), and speed and compression are bounded. Countability permits these assertions simultaneously for every pair of specified fields.

One concrete definition of a fiber is the linear span of the symbols e_j modulo the kernel of their Gram form. All finite Gram matrices have rank at most n, and some n-tuple is positive definite, so this fiber has dimension n. Continuity of Gram entries makes a positive-definite n-tuple a frame on an open time interval. Compactness of [a,b] gives a finite subdivision such that each closed piece lies in a frame interval and its inverse Gram matrix is bounded. Frame changes have W^{1,2} coefficients, as follows by multiplying cross-Gram entries by the inverse Gram matrix.

At almost every time these fibers are the original tangent fibers and the frame spans the tangent space. Thus \nabla_v e_i has L^2 coefficients in a frame, and (5.6) is meaningful. To verify that it is independent of a representation, suppose W=\sum_i a_ie_i is zero. Its pairing with each frame vector is identically zero. Differentiate that pairing using (5.7); the term pairing W with the derivative of the frame vector is zero almost everywhere. It follows that the right-hand side of (5.6) pairs to zero against a basis almost everywhere and hence vanishes. The same calculation proves metric compatibility. In each frame the parallel equation is a linear ODE with L^2, hence L^1, coefficients. Its absolutely continuous solutions exist uniquely. Metric compatibility preserves their pairings, and gluing at the subdivision endpoints constructs an orthonormal parallel frame on [a,b].

Parallel velocity. Put T=b-a. Apply [19, Theorem 5.13(i)] to the bounded-support optimal geodesic test plan after the affine reparameterization \widetilde\gamma_\tau=\gamma_{a+T\tau}. The functions g_i are in H^{2,2}, as required there. If \nabla\widetilde\varphi_\tau is its Kantorovich velocity, the first differentiation identity identifies the actual velocity in the original parameter as v_t=T^{-1}\nabla\widetilde\varphi_{(t-a)/T} along the plan almost everywhere. The second differentiation identity has a factor T^2 on both sides, giving, for e_i=\nabla g_i,

Its identity in L^2(\pi) yields the pathwise distributional identity by Fubini; its right-hand side is in L^2(dt\,d\pi) by bounded speed and compression. Equivalently one may apply the theorem to boundedly weighted restrictions of the plan, which remain optimal. Choose jointly measurable representatives u_i(t,\gamma)=\langle e_i,v\rangle(\gamma_t) and f_i(t,\gamma)=\Hess g_i(v,v)(\gamma_t) in L^2(dt\,d\pi). Their continuous scalar representatives can be specified by

This formula is jointly measurable. Along almost every path it belongs to W^{1,2}, has derivative f_i, and has the same average as u_i. It therefore represents the original scalar Sobolev class. Countability gives these assertions on one conull path set.

In a positive-Gram frame put c=P^{-1}u^*, where u^* is the vector of the corresponding scalar representatives. The coefficients c are W^{1,2}. Write \nabla_v e_i=\sum_j C_{ji}e_j. Metric compatibility and (5.8) give

Differentiation yields c'+Cc=0, proving \mathcal Dv=0 from the scalar Sobolev identities. Its almost-everywhere norm is the constant metric speed, so continuity gives the same norm at every time.

Measurable endpoint traces. The Gram identities defining the endpoint fibers agree with the actual tangent-module identities outside an m-null set. Endpoint compression excludes this set. Countability of the family makes the identification simultaneous for all specified fields. The endpoint vector can also be chosen measurably: enumerate the countable n-tuples and, at b, select the first tuple I=(i_1,\ldots,i_n) with positive Gram determinant. This defines a measurable countable partition of the good paths. In a measurable realization of the tangent module, set

This is measurable. The chosen minor remains positive on a neighborhood of b along that path, so the construction agrees with the continuous pathwise velocity just obtained. All its other specified gradient pairings equal u_j^*(b) by continuity, and its norm equals the path speed. The same construction applies at a. Thus the later endpoint L^2(\pi) pairings use a measurable bounded vector defined by scalar traces, independently of any choice of values for the original time-equivalence class.□

Proof. Choose a smooth scalar \eta with \eta(\delta)=0 and \eta(1)=1. For every interval with rational endpoints compactly contained in (\delta,1), choose a countable subset of C_c^\infty dense in its W^{1,2}_0 space. Let \mathcal A be their union and its rational linear span. Define \mathcal Y_{\delta,Z} to consist of

with arbitrary finite lists of indices. This is countable and has the required global endpoint identities.

Work on a path satisfying Proposition 5.6. Subtract \eta Z from W. In a global parallel frame the resulting field has W^{1,2}_0 coefficients. Cutting it off near \delta and 1 converges in W^{1,2}; this is the usual zero-trace approximation on an interval. A smooth temporal partition of unity then expresses the remaining compactly supported field in finitely many of the local test-gradient frames. Refine to intervals with rational endpoints whose closures remain in these frame intervals. The coefficient functions belong to W^{1,2}_0 on the corresponding intervals. Approximate them by elements of the chosen countable dense subsets. Such approximation converges uniformly as well as in W^{1,2} on an interval. Uniform convergence controls multiplication by the L^2 connection coefficients in (5.6). The fields therefore converge in the stated norm.□

Proof. Choose a compactly supported Lipschitz function \widetilde q equal to q on a neighborhood of \overline{B(p,R_\kappa)}, and compact test functions h_j\to\widetilde q in W^{1,2}. Include this deterministic countable sequence in the family of Proposition 5.1 before applying the pathwise statements. Test-function density here follows from heat regularization and good cutoffs; see [15].

Let \pi denote the radial restriction above, and fix one \delta<1. If (e_t)_\#\pi\leq C m on [\delta,1], then for any Lipschitz e\in W^{1,2} and 0<h<1-\delta, Sobolev differentiation and Jensen’s inequality give

For each fixed j, the continuous representative of \langle\nabla h_j,v\rangle gives

Dominated convergence applies with bound R_\kappa\operatorname{Lip}(h_j); no uniform bound in j is required. Terminal compression and |v(1)|\leq R_\kappa also imply

On the radial paths the quotient for \widetilde q is exactly (1-h/2)r(\gamma_1)^2. Apply (5.13) to e=\widetilde q-h_j, let h\downarrow0 first, and then let j\to\infty. This proves

Locality justifies replacing \widetilde q by q at the terminal points. Since |\nabla q|\leq r and |v(1)|=r(\gamma_1), equality in Cauchy–Schwarz proves (5.12), including the zero-speed case. The terminal measure is m on the ball, giving the last assertion.□

Proof. Use the radial data (5.11). For the fixed Z, adjoin its potential and the approximations in Lemma 5.9 to the countable family. Fix 0<\delta<1. For every trial (4.1) satisfying the global endpoint identities Y_\delta=0 and Y_1=Z, Theorem 4.1 gives

where f\geq0 is compactly supported and Lipschitz in the ball, the common calibrated path ending at x is denoted by \gamma^x, and

The common terminal prescription is essential here: the right-hand side is integration against a function of x, without averaging over several paths ending at x. The velocities agree under reweighting of these paths, since their pairings with test gradients are the derivatives of the same test functions along the paths. Bounded compression of \pi_\chi, bounded speed, and the L^2 covariant derivatives of each fixed trial show that E_Y\in L^1_{\mathrm{loc}}(B(p,R_\kappa),m).

We first extract a measure using one fixed trial, for example Y_t=(t-\delta)Z/(1-\delta), and denote its energy by E_0. The functional

is nonnegative on nonnegative compact Lipschitz functions. On a fixed compact subset, comparison with a Lipschitz cutoff equal to one there bounds its absolute value by a constant times \|f\|_\infty. Density in C_c and the Riesz representation theorem therefore give a positive Radon measure \nu representing it. Thus

is a signed Radon measure, with nonpositive singular part. Its representation of the fixed functional is unique, so the measure does not depend on the initially chosen trial or on \delta. Write \mu=h_Zm+\mu^{\mathrm s} for its Lebesgue decomposition.

Apply (5.15) to each member of the countable collection \mathcal Y_{\delta,Z} from Lemma 5.7. Each inequality says that E_Ym-\mu is a positive Radon measure. Consequently

The null set can be chosen simultaneously because the list is countable. The same is true for the pathwise calculus under \pi_\chi and its radial restriction.

On such a path, v is parallel and has norm r=r(x). Lemma 5.9 identifies its terminal direction with \nabla q(x). Parallel translate the two components Z_\parallel(x) and Z_\perp(x) to obtain parallel fields P_\parallel and P_\perp on [\delta,1]. For r>0 take the pathwise field

Its initial value is zero and its terminal value is Z(x). The denominator is positive by the choice of R_\kappa. At r=0 use the linear coefficient on the whole field. Lemma 5.7 approximates W in the pathwise norm by members of \mathcal Y_{\delta,Z}. The energy (5.16) is continuous in this norm because v is bounded. Hence the infimum in (5.17) is bounded above by the energy of W. Only this pathwise upper bound on a countable infimum is needed; no measurable selection of the fields W or their approximating sequences is required.

The parallel component contributes |Z_\parallel|^2/(1-\delta). If j(t)=\mathfrak s_\kappa((t-\delta)r)/ \mathfrak s_\kappa((1-\delta)r), then j''+\kappa r^2j=0, j(\delta)=0, and j(1)=1. Thus the transverse contribution is

We have proved

almost everywhere, with the continuous zero-speed value |Z|^2/(1-\delta). Choose a countable sequence \delta\downarrow0. The same density h_Z occurs for every \delta, so outside their common null set we may pass to the limit pointwise. Together with \mu^{\mathrm s}\leq0, this gives (5.14) as a measure inequality and therefore against every indicated Lipschitz weight.□

Proof. Write r=d(p,\cdot) and q=r^2/2, and fix 0<a<b<R_*. Choose a smooth nonincreasing function \theta\colon[0,\infty)\to[0,1], equal to one on [0,a^2/2] and zero on [b^2/2,\infty), and set

The quotient has a smooth extension at zero by its power series. Consequently F_p is bounded and globally Lipschitz, is constant outside B_b(p), and has the first value in (6.3). It need not lie in L^2(m); boundedness and a bounded gradient are the hypotheses needed for the weight.

For a compact test gradient Z=\nabla g, the definition (2.4) and the chain rule give

This identity involves only first derivatives of q, so it holds for the distributional Hessian. Local cutoffs keep its first term inside B_{R_*}(p). For r>0, put Z_r=\langle\nabla q,Z\rangle/r; then |Z_r|\le|Z|, since the Lipschitz bound for r gives |\nabla q|\le r almost everywhere. The Hessian comparison of Theorem 5.10 and \chi'\ge0 bound (6.4) above by

where, after cancellation of the model-distance factors,

The nonpositive singular part of H_q(Z,Z) remains nonpositive on multiplication by \chi'. Both coefficients extend continuously with A(0)=B(0)=1, and vanish for r\ge b. Since \theta'\le0 and \sk\ge0 on [0,b], we have A\le B: flattening the distance adds a nonpositive radial term. Hence the bounded continuous function

gives (6.1) globally. On r<a, it equals \ck(r)=1-\kappa F_p, as required.□

Proof. Theorems 4.1 and 5.10 supply the hypothesis of Lemma 6.2. Apply Theorem 6.1 to F_p,G_p. For every geodesic \sigma\colon[0,1]\to B_a(p) of length \ell, (6.3) yields

Consider a geodesic triangle [pxy] of perimeter less than a, and let \sigma be its chosen side from x to y. The triangle inequality along the two routes through the vertices gives d(p,\sigma_t)<a, so (6.5) holds on this entire side. The comparison triangle [\bar p\bar x\bar y] in the two-dimensional model of curvature \kappa exists at this scale. For its corresponding side \bar\sigma, the model cosine law gives

This also holds for degenerate triangles by continuity. Let h=y-\bar y. It belongs to H^1_0(0,1), and -h''-\kappa\ell^2h\ge0 in distributions. Testing with the negative part h_-, by nonnegative H^1_0 approximation, gives

If \kappa\le0, this forces h_-=0. If \kappa>0, our scale choice gives \kappa\ell^2<\pi^2, and the Dirichlet Poincaré inequality gives the same conclusion. Since \mathfrak m_\kappa is strictly increasing on the distances under consideration,

Thus every sufficiently short triangle, with any chosen sides, satisfies point-on-side comparison.

We spell out the localization for the final geometric step. Fix o\in M and choose r>0 with 20r<a. The local converse to point-on-side comparison gives the four-point \CBB(\kappa) inequality on B_r(o), as follows. Point-on-side comparison makes the comparison angle of a short hinge nonincreasing in either radial parameter. Its limit defines the angle, which is at least the comparison angle. The adjacent-angle inequality follows from the model gluing lemma. For a quadruple p,x,y,z\in B_r(o), take w\in(p,z) with d(p,w)<r. The radial legs from w to these points have length at most 3r, so every triangle used to compare points on these legs has perimeter at most 12r<a. Write \widetilde\angle_\kappa(w;x,y) for the comparison angle at w in the triangle wxy. For the actual angles at w, insert the direction wp between wx and wy in the angle triangle inequality. The resulting terms pair with wz into two adjacent-angle sums, each at most \pi. Each comparison angle is at most its actual angle, so

Letting w\to p, continuity of the comparison angles gives the same inequality with w replaced by p. This is the local four-point inequality, obtained by localizing the converse proof in [1, Theorem 8.14]. All segments here are ambient minimizing segments; no convexity of a metric ball is needed. Hence M is locally \CBB(\kappa) by the local four-point criterion in [1, Theorem 8.30, condition (1)]. Completeness and the length-space property allow the globalization theorem [1, Theorem 8.31], so M is \CBB(\kappa).

Finally, m=\mathcal H^n is finite on bounded sets and has full support. Every nonempty bounded ball therefore has positive finite n-dimensional Hausdorff measure, and hence Hausdorff dimension n. A countable exhaustion by such balls shows \dim_H M=n. This is the dimension of the resulting Alexandrov space.□

Proof of Theorem 1.1. If (A) holds, the Alexandrov-to-\RCD theorem gives \RCD((n-1)\kappa,n) with reference measure \mathcal H^n and full support. Theorem 3.1 gives the stated distributional sectional-curvature inequality, so (B) holds. Conversely, under (B), Theorem 4.1 gives the index inequality. Theorem 5.10 turns it into distributional Hessian comparison, and Theorem 6.3 yields the Alexandrov lower bound \kappa and dimension n. Thus (A) holds.□