## 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

### MSC:

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

Zbl 0503.53042
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### References:

 [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
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