In an influential paper, Li and Yau introduced the notion of conformal volume of a Riemannian manifold.

<cite authors="Li, Peter; Yau, Shing-Tung">_Li, Peter; Yau, Shing-Tung_, [**A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces**](http://dx.doi.org/10.1007/BF01399507), Invent. Math. 69, 269-291 (1982). [ZBL0503.53042](https://zbmath.org/?q=an:0503.53042).</cite>

See also [El Soufi and Ilias][1] for the generalization of the application to first eigenvalues in all dimensions. 



For a Riemannian manifold $M$, and $\phi: M\to S^n$ a (branched) conformal immersion, 
$$V_c(n, \phi) = \sup_{\gamma \in G} V(M,(\gamma\circ\phi)^* can),$$ where $G$ is the group of conformal (Möbius) transformations of $S^n$, and $can$ is the canonical round metric on $S^n$. Then $V_c(n,M) =\underset{ \phi:M\to S^n}{\inf} V_c(n,\phi)$, where the infimum is taken over all conformal immersions into $S^n$. Moreover, $V_c(M)=\lim_{n\to \infty} V_c(n,M)$. 

Then $V_c(M)$ is well-defined because of the Nash embedding theorem: there is an isometric embedding of $M$ into $\mathbb{R}^n$ for $n$ sufficiently large, and hence an conformal immersion into $S^n$. 


  [1]: https://doi.org/10.1007/BF01458460