Sobolev inequalities on manifolds: dependence of the constants on the Riemannian metric

Question

Let $g$ be a smooth Riemannian metric on the 2-torus $T^2$. $g$ induces the Sobolev space $W^{2,2}_g(T^2)$ via the norm $$ \|f\|_{W^{2,2}_g}^2 = \int_M |f|^2 + g(\nabla^2 f,\nabla^2 f)\, \text{vol}_g, $$ where $g$ is extended multi-linearly to all tensor bundles, $\nabla$ is the Levi-Civita connection of $g$, and $\text{vol}_g$ is the volume form. Since $g$ is equivalent to the flat metric on the torus, we have the Sobolev inequality $$ \|f\|_{L^\infty} \le C \|f\|_{W^{2,2}_g}. $$

Question: Is there any reference to the dependence of $C$ on intrinsic properties of $g$ (e.g., its volume and curvature)?

We are also interested in this question for other closed manifolds, and other Sobolev inequalities.

For example, when the underlying manifold is one dimensional, that is, $S^1$, then the only intrinsic property of the metric is the total length $\ell_g$, and one can get $$\|f\|_{L^{\infty}}^2 \leq \left(\ell_g/2+ 2/\ell_g\right) \|f\|_{W^{1,2}(g)}^2.$$ This is shown in Lemma~2.14 in the article by Bruveris-Michor-Mumford https://arxiv.org/pdf/1312.4995.pdf or, more generally, for open curves, Theorem 7.40 in Leoni's ``first course in Sobolev spaces,'' 2nd edition.

Answer 1

If you are interested in understanding arbitrary metrics $g$ on a 2-dimensional torus, you can proceed as follows. By the uniformatization theorem -- or equivalent simpler arguments -- we can write $g=\exp(2u) g_0$ where $g_0$ is a flat metric. It is not difficicult to do many such calculations explicitly for $g_0$. And one also sees: if you can control the function $u$ and its derivatives, then you can use Sobolev constants with respect to $g_0$ in order to get explicit, but in general not optimal Sobolev constants with respect to $g$.

It remains to control $u$ and its derivatives in terms of geometric data. A method called potential analysis may be used for this. I once worked out (as a PhD student without knowing that other people had done similar calculations) how to control the oscillation of $u$, i.e. $\mathrm{osc} u:= \mathrm{max} u- \mathrm{min} u$, see Section 3 of [Bernd Ammann, The Willmore Conjecture for immersed tori with small curvature integral, Manuscripta Math. 101, no. 1, 1-22 (2000), also available http://www.mathematik.uni-regensburg.de/ammann/preprints/willflat.html]. Probably, the derivatives of $u$ can be controlled similarly, but I have no precise reference at hand.