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.

  • $\begingroup$ I don't have time to look up the reference, but I recall that Taylor's three volumes on PDEs develops the theory over Riemannian manifolds and may have proofs that let you make dependence explicit. $\endgroup$ – Neal Oct 9 at 17:05
  • $\begingroup$ Try looking at the book Sobolev Spaces on Riemannian Manifolds by Hebey. $\endgroup$ – Deane Yang Oct 9 at 23:39
  • $\begingroup$ @DeaneYang : thanks, though it seems to me that Hebey's books only deal with subcritical Sobolev embeddings, and the ones I'm interested in are mostly super-critical (Morrey-type) inequalities. $\endgroup$ – C M Oct 10 at 12:49
  • $\begingroup$ You can probably get the $L^\infty$ bound from the sharp $L^2$ inequality using Moser iteration. My guess is that this is somewhere in a paper or book that requires elliptic estimates on a Riemannian manifold. If you know or learn the basic outline of what Moser iteration is, then you probably can work out the details yourself. You can see something like what you want in Appendix C of my paper Convergence of Riemannian manifolds with integral bounds on curvature. II. Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 2, 179–199. $\endgroup$ – Deane Yang Oct 10 at 18:41
  • $\begingroup$ @DeaneYang Thanks a lot! I'll look further in this direction. $\endgroup$ – C M Oct 10 at 18:57

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.

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  • $\begingroup$ Thanks for the idea and reference! $\endgroup$ – C M Oct 12 at 7:17

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