Upper bound for the first eigenvalue of algebraic submanifolds

Jean-Pierre Bourguignon Peter Li Shing-Tung Yau

Differential Geometry mathscidoc:1912.43517

Commentarii Mathematici Helvetici, 69, (2), 199-207, 1994.12
1. Statement of results Let Mn be a compact manifold of dimension n endowed with a Riemannian metric. The spectrum of the Laplacian,, acting on functions form a discrete set of the form {0< 1 2 k}. In 1970, Joseph Hersch [5] gave a sharp upper bound for the first non-zero eigenvalue 1 for any Riemannian metric on the 2-sphere in terms of its volume alone. Similar estimates for 1 on any compact oriented surfaces were derived by Yang-Yau [7]. The second and the third authors [6] studied the nonorientable surfaces and pointed out the relationship of 1 and the conformal class of the surface. In fact, their estimates were applied to study the Willmore problem. Another application of these types of upper bounds was found by Choi-Schoen [3] in relation to the set of all minimal surfaces in a compact 3-manifold of positive Ricci curvature. The purpose of this paper is to prove a higher dimensional generalization of the above results. It was pointed out by Marcel Berger [1] that Herschs theorem fails in higher dimensional spheres. In view of the relationship between 1 and the conformal
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  title={Upper bound for the first eigenvalue of algebraic submanifolds},
  author={Jean-Pierre Bourguignon, Peter Li, and Shing-Tung Yau},
  booktitle={Commentarii Mathematici Helvetici},
Jean-Pierre Bourguignon, Peter Li, and Shing-Tung Yau. Upper bound for the first eigenvalue of algebraic submanifolds. 1994. Vol. 69. In Commentarii Mathematici Helvetici. pp.199-207. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203747851434081.
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