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We show that the isoperimetric profile $h_{g(t)}(\xi)$ of a compact Riemannian manifold $(M,g)$ is jointly continuous when metrics $g(t)$ vary continuously. We also show that, when $M$ is a compact surface and $g(t)$ evolves under normalized Ricci flow, $h^2_{g(t)}(\xi)$ is uniform Lipschitz continuous and hence $h_{g(t)}(\xi)$ is uniform locally Lipschitz continuous.

Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow