Submanifolds with parallel mean curvature vector in a locally symmetric Riemannian manifold (Q2764450)

The idea of this paper is based on the famous works of minimal submanifolds on a sphere given by Simons and Chern- do Carmo-Kobayashi around 1970, i.e. to calculate \(\Delta S\) where \(S\) means the square of the norm of the second fundamental form of \(M\) in the sphere and \(\Delta\) means the Laplacian on \(M\). In this paper the author gives the following result: Let \(N^{n+p}\) be a locally symmetric complete Riemannian manifold with curvature pinching condition \(\frac 12<\delta\leq K_N\leq 1\), and \(M^n\) be a compact submanifold in \(N\) with parallel mean curvature vector field, denote \(H\) its mean curvature, \(S\) the square of norm of its second fundamental form, then we have the inequality NEWLINE\[NEWLINE\begin{multlined} \int\biggl\{ \bigl[+\frac 12\text{sgn} (p-1) \bigr]S^2- \bigl[n\delta +nH^2-H\sqrt{n(n-1) (S-nH^2)}- \frac 13 n(p-1)(1-\delta) \bigr]\\ S+n^2\delta H^2+ \frac{1}{72} n(n-1)(26 n-25)p(1-\delta)^2 \biggr\} \ast 1\geq 0,\end{multlined}NEWLINE\]NEWLINE where \(\text{sgn}\) means sign function and \(\ast 1\) volume element of \(M\). When \(\delta=1\) and \(H=0\) the inequality implies Simons' inequality NEWLINE\[NEWLINE\int S\Bigl(S-\frac {n}{2-\frac 1p} \Bigr)\ast 1\geq 0.NEWLINE\]NEWLINE As a corollary of the main result, if the integrand \(\leq 0\), then \(N^{n+p}\) has to be \(N^{n+p}(1)\), the space of constant curvature 1, and \(M\) is of parallel second fundamental form.