I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of Optimal Transport Theory (As far as I know, it hardly is)

The observation is as follows (which has been validated with some toy simulations).

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Problem (not rigorously stated)

Let $\mu, \nu$ probabilistic measures uniformly on regular manifolds $\mathcal{M}, \mathcal{N}$, $C^{\infty}(\mathcal{M}, \mathcal{N})$ the set of continuous mapping from $\mathcal{M}$ to $\mathcal{N}$, and $\Pi(\mu,\nu)$ the set of measures on $\mathcal{M}\times\mathcal{N}$ s.t. its marginal distributions are respectively $\mu, \nu$.

Consider the following optimization problem

$$ (*) = \min_{T\in{C^{\infty}(\mathcal{M}, \mathcal{N})}} \inf_{\gamma\in\Pi(\mu,\nu)} \int c(T(x), y)d\gamma(x,y) $$ where $c(\cdot, \cdot)$ is a cost function defined on $\mathcal{N}\times\mathcal{N}$.

My question is that whether $(*) \propto h(\chi(\mathcal{M}), \chi(\mathcal{N}))$, or generally, the difference between the manifolds' topological invariants.

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Look forward to any feedbacks and welcome discussions :D. I would also provide the details of toy experiments if anyone is interested in.