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.
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2 | changed \varphi^c to \x\mapsto x+\nabla\varphi^c | ||
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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. |
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