New characterizations of the Clifford tori and the Veronese surface.(English)Zbl 0791.53056

Let $$M$$ be an $$n$$-dimensional compact minimal (immersed) submanifold of the unit sphere $$S^{n+p}(1)$$. Let $$S$$ denote the length of the second fundamental form of $$M$$. It is known that, if $$S\leq n/(2-1/p))$$, then $$S=0$$ or $$S=n/ (2-1/p))$$ [J. Simons, Bull. Am. Math. Soc. 73, 491- 495 (1967; Zbl 0153.232)]; in the latter case, $$M$$ is the Veronese surface in $$S^ 4(1)$$ or a Clifford torus in $$S^{n+1}(1)$$ [S. S. Chern, M. P. do Carmo and S. Kobayashi, Functional analysis related fields, Conf. Chicago 1968, 59-75 (1970; Zbl 0216.440)]. Further it is known that if $$p\geq 2$$ and $$S\leq 2n/3$$, then either $$S=0$$ or $$M$$ is the Veronese surface in $$S^ 4(1)$$ [A. Li and J. Li, Arch. Math. 58, 582-594 (1992; Zbl 0731.53056)]. Here the author introduces two Schrödinger operators $$L_ 2$$ and $$L_ 3$$ by $$L_ 2=- \Delta- (2-1/p)S$$ and $$L_ 3=- \Delta -3/2 S$$ and studies the first eigenvalues $$\mu_ 1$$ of $$L_ 2$$ and $$\sigma_ 1$$ of $$L_ 3$$. The main results are: (1) either $$\mu_ 1=0$$ and $$M$$ is totally geodesic or $$\mu_ 1\leq -n$$; if $$\mu_ 1=-n$$, then $$M$$ is the Veronese surface in $$S^ 4(1)$$ or a Clifford torus in $$S^{n+1}(1)$$; (2) either $$\sigma_ 1=0$$ and $$M$$ is totally geodesic or $$\sigma_ 1\leq-n$$; if $$\sigma_ 1=-n$$, then $$M$$ is the Veronese surface in $$S^ 4(1)$$.
Reviewer: F.Dillen (Leuven)

MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text:

References:

 [1] K. Benko, M. Kothe, K. D. Semmler andU. Simon, Eigenvalue of the Lapiacian and curvature. Colloq. Math.42, 19-31 (1979). · Zbl 0437.53032 [2] S. S.Chern, Minimal submanifolds in a Riemannian manifold. Kansas 1968. [3] S. S.Chern, M.Do Carmo and S.Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length. In: Selected Papers, S. S. Chern ed., 393-409, Berlin-Heidelberg-New York 1978. [4] A.-M. Li andJ. M. Li, An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math.58, 582-594 (1992). · Zbl 0767.53042 [5] Y. B. Shen, Curvature and stability for minimal submanifolds. Sci. Sinica Ser. A31, 787-797 (1988). · Zbl 0658.53055 [6] J. Simons, Minimal varieties in Riemannian manifolds. Ann. of Math. (2)88, 62-105 (1968). · Zbl 0181.49702 [7] C. X. Wu, A characterization of Clifford minimal hypersurfaces (Chinese). Adv. in Math. (Beijing)18, 352-355 (1989). · Zbl 0694.53056
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.