[proofplan]
The direct smooth-map Jacobian calculation proves the result only before cut-locus and nonsingularity issues enter, so we use the standard global theorem that packages exactly those analytic difficulties. The weighted Riemannian curvature-displacement convexity theorem of von Renesse-Sturm, equivalently the smooth weighted case of the Lott-Sturm-Villani curvature-dimension theorem with dimension parameter $\infty$, says that on a complete weighted Riemannian manifold the lower bound $\operatorname{Ric}+\operatorname{Hess}_gV\ge Kg$ implies $K$-convexity of the relative entropy along some optimal $W_2$-geodesic between any two finite-entropy endpoints. We state the precise form of that external theorem, verify its hypotheses from the assumptions here, and then identify its conclusion with the desired inequality.
[/proofplan]
[step:State the global weighted displacement convexity theorem]
We use the following standard external theorem, often called the von Renesse-Sturm weighted Riemannian characterization of lower Ricci bounds or the smooth weighted Lott-Sturm-Villani theorem.
[claim:Weighted Riemannian displacement convexity theorem]
Let $(N,h)$ be a complete connected smooth Riemannian manifold without boundary, let $W\in C^\infty(N;\mathbb R)$, and let $q$ be the Borel measure defined by $dq=e^{-W}\,d\operatorname{vol}_h$. Assume that $q$ is locally finite and has full support. Let $\mathcal P_2(N)$ be the quadratic Wasserstein space induced by the Riemannian distance of $h$. Let $C([0,1];N)$ carry its compact-open topology, and let $e_s:C([0,1];N)\to N$ be the evaluation map $e_s(\gamma)=\gamma(s)$. For $\beta\in\mathcal P_2(N)$, let $\operatorname{Ent}_q(\beta)$ denote relative entropy with respect to $q$.
Let $L\in\mathbb R$, and assume that $\operatorname{Ric}_h+\operatorname{Hess}_hW\ge Lh$ as quadratic forms on $TN$. If $\alpha_0,\alpha_1\in\mathcal P_2(N)$ have finite real relative entropy with respect to $q$, then there exists an optimal dynamical plan $\Gamma\in\mathcal P(C([0,1];N))$ concentrated on constant-speed minimizing $h$-geodesics such that, defining $\alpha_s:=(e_s)_\#\Gamma$ for $s\in[0,1]$, the curve $s\mapsto\alpha_s$ is a constant-speed $W_2$-geodesic from $\alpha_0$ to $\alpha_1$ and satisfies
\begin{align*}
\operatorname{Ent}_q(\alpha_s)\le (1-s)\operatorname{Ent}_q(\alpha_0)+s\operatorname{Ent}_q(\alpha_1)-\frac{L}{2}s(1-s)W_2^2(\alpha_0,\alpha_1)
\end{align*}
for every $s\in[0,1]$.
[/claim]
[proof]
This is the standard global theorem of von Renesse and Sturm, refined in the Lott-Sturm-Villani theory for smooth weighted manifolds. Its proof consists of the smooth Jacobian second-variation formula for displacement interpolations, McCann's approximation through regular transport maps, entropy-recovery approximation of finite-entropy measures in $W_2$, stability of optimal dynamical plans, and lower semicontinuity of relative entropy for locally finite sigma-finite reference measures. We use it as an external theorem in precisely the form stated above.
[/proof]
[guided]
The point of isolating this theorem is to avoid hiding the hard part of the argument inside phrases such as regularization or stability. The local computation along a nonsingular optimal map is only a model calculation. The global theorem stated here is the standard result that upgrades that model calculation to arbitrary finite-entropy endpoints on a complete weighted manifold.
The theorem has the following data. The ambient space is a complete connected smooth Riemannian manifold $(N,h)$ without boundary. The reference measure is a smooth weighted measure $q$ given by $dq=e^{-W}\,d\operatorname{vol}_h$, where $W:N\to\mathbb R$ is smooth. The measure hypotheses are local finiteness and full support. The curvature hypothesis is the tensor inequality $\operatorname{Ric}_h+\operatorname{Hess}_hW\ge Lh$. The conclusion is not merely an infinitesimal convexity statement: for each pair of finite-entropy endpoints $\alpha_0,\alpha_1\in\mathcal P_2(N)$, it produces an optimal dynamical plan $\Gamma$ on the path space $C([0,1];N)$.
Here an optimal dynamical plan means a Borel probability measure on $C([0,1];N)$ which is concentrated on constant-speed minimizing geodesics and whose endpoint coupling $(e_0,e_1)_\#\Gamma$ is optimal for the quadratic transport cost. The evaluation map $e_s:C([0,1];N)\to N$ is defined by $e_s(\gamma)=\gamma(s)$. The time marginal is therefore $\alpha_s=(e_s)_\#\Gamma$. The theorem asserts that these marginals form a constant-speed $W_2$-geodesic and satisfy the displayed entropy inequality at every time $s\in[0,1]$.
This theorem is the exact external input needed here. It contains the Riccati-Jacobian second variation in the smooth regular case and also contains the global approximation and lower semicontinuity arguments required to pass through cut-locus singularities, nonsmooth optimal maps, noncompactness, and merely finite entropy endpoints.
[/guided]
[/step]
[step:Verify the hypotheses of the global theorem]
Apply the preceding theorem with
\begin{align*}
(N,h,W,q,L)=(M,g,V,m,K)
\end{align*}
The manifold hypotheses hold because $(M,g)$ is assumed complete, connected, smooth, and without boundary. The weight hypothesis holds because $V\in C^\infty(M;\mathbb R)$ and $dm=e^{-V}\,d\operatorname{vol}_g$. The measure hypotheses hold because the theorem assumes that $m$ is locally finite and has full support.
The curvature hypothesis of the external theorem is exactly the stated Bakry-Emery lower bound. Indeed, for every $x\in M$ and every $\xi\in T_xM$,
\begin{align*}
(\operatorname{Ric}+\operatorname{Hess}_gV)_x(\xi,\xi)=\operatorname{Ric}_V|_x(\xi,\xi)\ge K g_x(\xi,\xi)
\end{align*}
Thus $\operatorname{Ric}+\operatorname{Hess}_gV\ge Kg$ as quadratic forms on $TM$.
Finally, the endpoint hypotheses of the external theorem hold by assumption: $\mu_0,\mu_1\in\mathcal P_2(M)$ and both $\operatorname{Ent}_m(\mu_0)$ and $\operatorname{Ent}_m(\mu_1)$ are finite real numbers.
[/step]
[step:Extract the entropy convex geodesic]
By the weighted Riemannian displacement convexity theorem, there exists an optimal dynamical plan $\Pi\in\mathcal P(C([0,1];M))$ concentrated on constant-speed minimizing $g$-geodesics such that $(e_0,e_1)_\#\Pi$ is an optimal coupling of $\mu_0$ and $\mu_1$. For each $t\in[0,1]$, define the time marginal
\begin{align*}
\mu_t:=(e_t)_\#\Pi
\end{align*}
where $e_t:C([0,1];M)\to M$ is the evaluation map $e_t(\gamma)=\gamma(t)$.
The same theorem gives that
\begin{align*}
\mu:[0,1]\to\mathcal P_2(M),\qquad t\mapsto\mu_t
\end{align*}
is a constant-speed $W_2$-geodesic from $\mu_0$ to $\mu_1$. It also gives, for every $t\in[0,1]$,
\begin{align*}
\operatorname{Ent}_m(\mu_t)\le (1-t)\operatorname{Ent}_m(\mu_0)+t\operatorname{Ent}_m(\mu_1)-\frac{K}{2}t(1-t)W_2^2(\mu_0,\mu_1)
\end{align*}
This is precisely the asserted $K$-convexity inequality.
[/step]
[step:Identify the weak optimal-plan conclusion]
The construction above gives more than a curve: it gives an optimal dynamical plan $\Pi$ whose time marginals are the measures $\mu_t$. Since $\Pi$ is concentrated on constant-speed minimizing geodesics and its endpoint coupling is optimal, it is an optimal plan in the dynamical sense. Therefore the entropy inequality holds along the time marginals of at least one optimal dynamical plan connecting the endpoints.
This is exactly the weak optimal-plan form of $K$-displacement convexity on the finite-entropy domain of $\operatorname{Ent}_m$. The theorem follows.
[/step]
