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In the paper [1] by Bianchi and Egnell, they proved the following discrepancy result for the classical Sobolev embedding $$ \|\nabla\phi\|_{2}^2-S^2\|\phi\|_{2^*}^2\geq\alpha d(\phi,M)^2. $$ This result holds in an Euclidean set up. Are there analogous results for standard manifolds like spheres or hyperbolic manifolds?
Can anyone suggest some references where this kind of results have been shown in manifolds other than $\Bbb R^n$?

Reference

[1] Gabriele Bianchi, Henrik Egnell, "A note on the Sobolev inequality", (English) Journal of Functional Analysis 100, No. 1, 18-24 (1991), MR1124290, Zbl 0755.46014.

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  • $\begingroup$ One natural formulation of this on manifolds is in the context of the Yamabe problem; see arxiv.org/pdf/2009.14362.pdf for example. $\endgroup$
    – user378654
    Commented Feb 8, 2023 at 4:46
  • $\begingroup$ Thanks. Sure, I will check that out. $\endgroup$
    – User1723
    Commented Feb 8, 2023 at 5:53

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