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Immersions minimales, première valeur propre du laplacien et volume conforme. (Minimal immersions, first eigenvalue of the Laplacian and conformal volume). (French) Zbl 0675.53045
The authors study the conformal volume \(V_ c(M)\) defined for a conformal class of metrics on a compact manifold by P. Li and 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.
Reviewer: G.Thorbergsson

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
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