Minimizing movement scheme for intrinsic aggregation on compact Riemannian manifolds

The paper proves small-time existence of measure-valued solutions to the intrinsic aggregation equation on a compact Riemannian manifold by a minimizing-movement (JKO) construction, deriving a discrete Euler–Lagrange identity, a finite-speed propagation bound spt(μ_{k+1}) ⊂ N_{Lτ}(spt(μ_k)), and passing to the limit in the weak formulation; crucially, it ensures the supports remain away from the cut locus up to time T < δ_{μ0}/(2L), establishing differentiability where needed (Theorem 4, Proposition 17, Lemmas 10 and 16, and the limit evaluation around equation (27)) . The candidate solution follows the same JKO framework and discrete EL identity, obtains the same finite-speed and δ(t) ≥ δ_{μ0} − 2Lt bound, and then passes to the weak limit via a commutator estimate exploiting uniform away-from-cut regularity of ∇W and its convolution. The main difference is technical: the paper passes to the limit using a product-measure contraction estimate and Kantorovich–Rubinstein duality, while the model uses a quantitative “commutator” bound; both are consistent under the paper’s hypotheses (W0)–(W1) . A minor caveat in the model is that it does not explicitly justify differentiability of the c-transform on spt(μ_{k+1}); the paper supplies this via semiconcavity and the EL condition. Overall, the conclusions and time-of-existence threshold agree.