New characterizations of the Clifford tori and the Veronese surface (Q689833)

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 \(S=n/ (2-1/p))\) [\textit{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 \(S^{n+1}(1)\) [\textit{S. S. Chern}, \textit{M. P. do Carmo} and \textit{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 \(S^ 4(1)\) [\textit{A. Li} and \textit{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 \(S^ 4(1)\).