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

Li, Peter; Yau, Shing-Tung, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69, 269-291 (1982). ZBL0503.53042.

See also El Soufi and Ilias 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 $S^n$.

El Soufi and Ilias prove the theorem:

Theorem Let $(M,g)$ be a compact Riemannian manifold of dimension $m$. Then $$\lambda_1(M,g) V(M,g)^{2/m}\leq V_c(M)^{2/m}.$$

Equality holds iff $(M,g)$ admits an isometric immersion into $S^n$ (up to scaling) by first eigenfunctions.

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

Li, Peter; Yau, Shing-Tung, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69, 269-291 (1982). ZBL0503.53042, MR674407, eudml.

See also El Soufi and Ilias 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 $S^n$.

El Soufi and Ilias prove the theorem:

Theorem Let $(M,g)$ be a compact Riemannian manifold of dimension $m$. Then $$\lambda_1(M,g) V(M,g)^{2/m}\leq V_c(M)^{2/m}.$$

Equality holds iff $(M,g)$ admits an isometric immersion into $S^n$ (up to scaling) by first eigenfunctions.

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

Li, Peter; Yau, Shing-Tung, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69, 269-291 (1982). ZBL0503.53042.

See also El Soufi and Ilias 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 $S^n$.

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

Li, Peter; Yau, Shing-Tung, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69, 269-291 (1982). ZBL0503.53042.

See also El Soufi and Ilias 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 $S^n$.

El Soufi and Ilias prove the theorem:

Theorem Let $(M,g)$ be a compact Riemannian manifold of dimension $m$. Then $$\lambda_1(M,g) V(M,g)^{2/m}\leq V_c(M)^{2/m}.$$

Equality holds iff $(M,g)$ admits an isometric immersion into $S^n$ (up to scaling) by first eigenfunctions.