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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.)
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References:

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