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Concerning (2), if I don't mix up notations and when $\nu$ is absolutely continuous, $\varphi^c$ x\mapsto x+\nabla\varphi^c$is the Brenier map from$\nu$to$vol_M$. As a consequence, it holds$d^W(vol_M,\nu)=\int_M|\nabla\varphi^c|_g^2 d\nu$, so if$\nu$is far away from$vol_M$and$\varphi$is not too close to be constant, then your two integrals can be very different, and I do not see why they should have a relation. 1 Concerning (2), if I don't mix up notations and when$\nu$is absolutely continuous,$\varphi^c$is the Brenier map from$\nu$to$vol_M$. As a consequence, it holds$d^W(vol_M,\nu)=\int_M|\nabla\varphi^c|_g^2 d\nu$, so if$\nu$is far away from$vol_M$and$\varphi\$ is not too close to be constant, then your two integrals can be very different, and I do not see why they should have a relation.