Immersions minimales, première valeur propre du laplacien et volume conforme. (Minimal immersions, first eigenvalue of the Laplacian and conformal volume) (Q1122161)

The authors study the conformal volume \(V_ c(M)\) defined for a conformal class of metrics on a compact manifold by \textit{P. Li} and \textit{S.-T. Yau} [Invent. Math. 69, 269-291 (1982; Zbl 0503.53042)]. They first show that the volume of a minimal submanifold in a sphere is greater or equal to its conformal volume. This was proved by Li and Yau for minimal surfaces. Then they prove that \(\lambda_ 1(M,g)V(M,g)^{2/m},\) where m is the dimension of M, is bounded from above by \(mV_ c(M)^{2/m}.\) This was proved in dimension two by Li and Yau. They remark that it follows from the work of some authors that there is a family \(g_ t\) of metrics on \(S^ m\), \(m\geq 3\), such that \(\lambda_ 1(S^ m,g_ t)V(S^ m,g_ t)^{2/m}\) tends to infinity with t. Finally they study Riemannian manifolds M such that their conformal class of metrics contains a metric \(g_ 0\) that can be minimally immersed into a sphere by first eigenvalue functions. It follows that \(V(M,g_ 0)=V_ c(M)\) and \(\lambda_ 1(M,g)V(M,g)^{2/m}\leq \lambda_ 1(M,g_ 0)V(M,g_ 0)^{2/m}.\) As an application of the second inequality they show that a conformal class of metrics cannot have more than one metric that admits minimal immersions into a sphere by first eigenfunctions.