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)


Zbl 0503.53042
Full Text: DOI EuDML


[1] Beardon, A.F.: The geometry of discrete groups. Graduate texts in mathematics, 91. Berlin, Heidelberg, New York: Springer 1983
[2] Berard-Bergery, Bourguignon, J.P.: Laplacian and riemannian submersions with totally geodesic fibres. Ill. J. Math.26, 181-200 (1982) · Zbl 0483.58021
[3] Berger, M.: Sur les premières valeurs propres des variétés riemanniennes. Compos. Math.26, 129-149 (1973) · Zbl 0257.53048
[4] Besse, A.L.: Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete 93. Berlin, Heidelberg, New York: Springer 1978 · Zbl 0387.53010
[5] Bourguignon, J.P.: Première valeur propre du laplacien et volume des sphères riemanniennes. Sémin. Goulaouic-Schwartz Equations Deriv. Partielles9, 1-17, 1979-1980
[6] Dieudonne, J.: Eléments d’analyse, tome IX. Paris: Gauthier-Villars 1982
[7] El Soufi, A., Ilias, S.: Le volume conforme et ses applications d’après Li et Yau. Publications du séminaire de théorie spectrale et géométrie. Chambéry-Greoble, exposé VII, 1-15, (1983-1984)
[8] Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris270, 1645-1648 (1970) · Zbl 0224.73083
[9] Lawson, H.B.: Lectures on minimal submanifolds. Lecture series 9. Berkeley: Publish or Perish 1980 · Zbl 0434.53006
[10] Li, P., Yau, S.T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math.69, 269-291 (1982) · Zbl 0503.53042
[11] Montiel, S., Ros, A.: Minimal immersions of surfaces by the first eigenfunctions and conformal area. Preprint 1985 · Zbl 0584.53026
[12] Muto, H.: The first eigenvalue of the laplacian on even dimensional spheres. Tôhoku Math. J.32, 427-432 (1980) · Zbl 0435.53035
[13] Muto, H., Ohnita, Y., Urakawa, H.: Homogeneous minimal hypersurfaces in the unit sphere and the first eigenvalue of their laplacian. Tôhoku Math. J.36, 253-267 (1984) · Zbl 0539.53044
[14] Tanno, S.: The first eigenvalue of the laplacian on spheres. Tôhoku Math. J.31, 179-185 (1979) · Zbl 0403.53019
[15] Urakawa, H.: On the least positive eigenvalue of the lapacian for compact group manifold. J. Math. Soc. Japan31, 209-226 (1979) · Zbl 0402.58012
[16] Yang, P., Yau, S.T.: Eigenvalues of the laplacian of compact rieman surfaces and minimal submanifolds. Ann. Scuola Sup. Pisa7, 55-63 (1980) · Zbl 0446.58017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.