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.