Metric-Dependent Nonlocal Entropy and Coupled WassersteinRiemannian Gradient Dynamics

Model and entropy--------------------Let \(M\) be a closed connected manifold, \(gref\) a reference metric, and \(ρ_g=dμ/dvol_g\). The energy is \[ F[μ,g]= E[μ,g]+λ R[g]+κ S[g], R[g]=∫_M R_g\,dvol_g, S[g]=\|g-gref\|_{H^s(gref)}^2.\] \[ E[μ,g]= Kε,g(x,y)logρ_g(y)\,dμ(x)dμ(y),∫_M Kε,g(x,y)\,dvol_g(y)=1.\] Kernels are smooth, nonnegative, row-Markov and covariant. Examples are the heat kernel and a normalized distance kernel in a uniform normal neighbourhood. Row normalization yields \( E≥-1/e\). The metric enters through kernel, density and volume. At fixed measure, volume variations cancel against \(δ_hρ=-12tr_g(h)ρ\). 2. First variations and moving-volume identities------------------------------------------------The measure potential, up to an additive constant on zero-mass variations, is \[Φ= K_g(ρlogρ)+( K_g^*ρ)(1+logρ).\] The metric differential at fixed \(μ\) is \[D_g E[h]=∫_MΨ,h_gdvol_g+ QK,g(μ,h), Ψ=-12( K_g^*ρ)ρ g,\]\[ QK,g(μ,h)=(δ_hKε,g)(x,y)logρ(y)\,dμ(x)dμ(y).\] For \(P_t^g=etΔ_g\), \[D_gP_t^g[h]=∫_0^tPₜ₋ᵣ^g(D_gΔ_g[h])P_r^gdr, D_gΔ_g[h]f=-hⁱʲ∇_i∇_jf-_g h-12dtr_g h,df_g,\]\[kD_gP_t^g[h](x,y)=δ_hpt,g(x,y)+12tr_g(h)(y)pt,g(x,y).\] For this distance kernel, the appendices derive \(δ_hd_g(x,y)^2=∫_0^1h_{γxy(r)}(γ,γ)dr\) and \(δ_hZ=∫_Mδ_h K\,dvol_g+12∫_M Ktr_g(h)dvol_g\); and the quotient rule gives \(δ_hK=(δ_h K)/Z- K\,δ_hZ/Z^2\). For the heat kernel, put \(f=ρlogρ\), \(f_v=Pτ-v^gf\), and \(r_v=P_v^gρ\). Then \[ M[h]=∫_M T_τ,h_gdvol_g,_τ=-12ρ(P_τ^gρ+P_τ^gf)g+∫_0^τ\![df_v dr_v-12( df_v,dr_v_g+f_vΔ_gr_v)g]dv.\] The tensor is Lipschitz in \(L^2(gref)\) on controlled positive trace sets; the metric gradient is \(C^1\) by the \(C^2\) theorem and Riesz map. 3. Covariance, DeTurck gauge and energy---------------------------------------For \(φ_#g=(φ⁻¹)^*g\), \[ E[φ_#μ,φ_#g]= E[μ,g], R[φ_#g]= R[g],_g E[ L_Vg]=D_μ E[-div_g(μ V)]=∫_M∇_gΦ,V_gdμ.\] The penalty \( S\) is not generally covariant. For \(W^k=gⁱʲ(Γ^kᵢⱼ(g)-Γ^kᵢⱼ(gref))\), partial DeTurck modifies only the metric equation by \( L_Wg\); the full version also adds \(div_g(μ W)\) to the measure equation. For \( D_μ=∫_M|∇_gΦ|^2dμ\) and \( D_g=η\|GradH^s_g F\|H^s^2\), the partial balance is \[d F/dt=- D_μ- D_g+ CDeT, CDeT=∫_M∇_gΦ,W_gdμ+2κ g-gref, L_WgH^s.\] The full balance is \(d F/dt=- D_μ- D_g+2κ g-gref, L_WgH^s\). At \(κ=0\), smooth formal solutions satisfy \( F(T)+∫_0^T( D_μ+ D_g)dt= F(0)\); this is not a well-posedness result. Under an evolving metric, \[∂_tρ+12_g(∂_tg)=div_g(ρ∇_gΦ),\] with an additional \(div_g(ρ W)\) in the full system. 4. Proven analytic scope------------------------For fixed \(τ=ε^2>0\), the entropy is proved \(C^2\) on positive \(H^m× H^m\), \(m>d/2+4\), with quadratic remainder and locally Lipschitz differential. It extends to the trace space \(Hˢ⁻¹× H^s\), integer \(s>d/2+6\). No bound is uniform as \(τ↓0\). The local well-posedness theorem is only for the heat kernel, fixed \(τ>0\), \(λ=0\), \(κ>0\), and no DeTurck term. In fixed reference-volume coordinates, let \(q\,dvol_{gref}=μ\), \(J_g=dvol_g/dvol_{gref}\), \(ρ=q/J_g\), \(A=P_τ^gρ\), and \(B=P_τ^g(ρlogρ)\). The subsystem is \[∂_tq=J_gdiv_g(A∇_gρ+ρ∇_gB+ρ(1+logρ)∇_gA),∂_tg=-H^s_g F_0, F_0= E_τ+κ\|g-gref\|H^s^2.\] The principal symbol is \(σ_2( A)(x,ξ)=A(x)gⁱʲξ_iξ_j\), with \(A≥ρ>0\). For positive unit-mass data in \(Hˢ⁻¹× H^s\), the theorem proves local maximal-regularity existence and uniqueness, positivity persistence, mass conservation, locally Lipschitz dependence and \[ F_0(t)+∫_0^t\!∫_M|∇_gΦ|^2dμ dr+η∫_0^t\|GradH^s_g F_0\|H^s^2dr= F_0(0).\] It gives no well-posedness theorem for the full model, global existence or zero-heat-time limit. 5. JKO splitting and conditional errors---------------------------------------The measure step minimizes \(1{2h}W2,g_n^2(ν,μ_n)+ F[ν,g_n]\) over a local admissible set; the metric is then updated by \(gₙ₊₁=g_n+h[-H^s F[μn+1/2,g_n]+ LW(g_n)g_n]\). Under an interior-minimizer assumption and local Lipschitz control, the one-step energy inequality contains the decrease \(-W_2^2/(2h)\), metric descent, the DeTurck term and an \(O(h^2)\) metric remainder. The continuous residual is \(O(h^2)\) in the metric norm and against \(C^2\) tests. The minimizer-level comparison is conditional on a \(C^2\) Wasserstein chart, controlled exponential/logarithm maps, unique regular optimal maps avoiding the cut locus, an interior unique minimizer, and a \(C^1\) charted frozen field with bounded second derivative. The coupled estimate additionally assumes an existing smooth solution with \(g∈ C^1(I_n;Hˢ⁺²)∩ C^2(I_n;H^s)\), measure-Lipschitz metric field, and metric-Lipschitz charted measure field in \(Hˢ⁺²\). Under these assumptions only, \[W2,g_n(μ_h^J,ν_n(h))≤ C_nh^2, W2,g_n(μₙ₊₁,μ(t_n+h))≤ C_nh^2,\| gₙ₊₁-g(t_n+h)\|H^s≤ C_nh^2.\] These are local exact-state bounds, not global convergence; the well-posedness theorem supplies neither the extra regularity nor the transport chart. Without a smooth chart, the fallback is test-function \(o(h)\) consistency, or conditional \(O(h^2)\) with stronger quantitative assumptions. Passage to weak equations is conditional on unproved compactness assumptions (S1)–(S4). 6. Explicit circle checks-------------------------For \(g_a=a^2dθ^2\) and \(dμ_δ=(1+δcosθ)dθ/(2π)\), set \(c_a=(2π a)⁻¹\), \(L_a=log c_a\), \(β_a=e-τ/a^2\), and \(A_a=1+2β_a(1+L_a)\). Then \[ E(a,δ)=c_aL_a+c_aA_a/4δ^2+O(δ^4), A_a'=4τβ_a/a^3(1+L_a)-2β_a/a.\] The first-mode linearization is \(δ=-A_aδ/(2π a^3)+O(δ^2)\); along a changing homogeneous branch, its linear solution is \(δ(t)=δ(0)exp[-∫_0^t Aₐ₍ᵣ₎/(2π a(r)^3)\,dr]\). For a general fixed metric, if \(-Δ_ge_j=λ_je_j\), then \[DF_τ(c)u=cΔ_g(u+2(1+log c)P_τ u),_τ(c)e_j=-cλ_j[1+2(1+log c)e-τλ_j]e_j.\] For \(τ>0\), all zero-mass modes decay iff \(1+2(1+log c)e-τλ_1>0\); otherwise finitely many grow. At \(τ=0\), nonconstant modes decay for \(c>e-3/2\), are neutral at equality and grow unboundedly below it. On the circle, \(W(g_a)=0\), \[\|∂_ag_a\|_{H^s(gref)}^2=8π a^2/a_0^3, S[g_a]=2π(a^2-a_0^2)^2/a_0^3,\]\[ a=-η[a_0^3/8π a^2∂_a E+^2-a_0^2/a].\] On the homogeneous branch \(δ=0\), this reduction is exact and \( a=η a_0^3(1+log(1/(2π a)))/(16π^2a^4)-ηκ(a^2-a_0^2)/a\). For \(δ≠0\), one Fourier mode is not invariant and the metric equation is only projected onto \(g_a\).