Eigenvalue inequalities and minimal submanifolds (Q1392774)

The author studies the relationship between the eigenvalues of the Laplace operator and the mean curvature vector field \(H\) for an \(n\)-dimensional manifold \(M\) isometrically immersed into the \(m\)-dimensional sphere \(S^m\). Let \(x: M \to S^n \to\mathbb{R}^{n+1}\) be the composition of such an immersion with the inclusion of \(S^n\) in \(\mathbb{R}^{n+1}\). Then \(x\) has an \(L^2\)-orthogonal decomposition \(x = x_0 + \sum_{u \geq 1} x_u\) into \(\mathbb{R}^{n+1}\)-valued eigenfunctions of the Laplacian on \(M\). If \(x_0 = 0\), then the immersion \(x\) is called mass-symmetric. If the sum is only over a finite set of indices \(u_i\), \(i=1,\dots,k\), then \(x\) is said to be of \(k\)-type and to have order equal to \(\{ u_1, \dots , u_k \}\). One of the results in the paper is that if \(x\) is mass-symmetric and of order \(\{k,k+1\}\) for some \(k\) such that \(\lambda_k \geq n\) or \(\lambda_{k+1} \leq n\), then the immersion is minimal and \(\lambda_k = n\) or \(\lambda_{k+1} = n\). Also, the author proves the inequality \(\lambda_1 \leq nV^2/(V^2 - (\int H)^2)\). Especially, if \(x\) is mass-symmetric, then \(\lambda_1 \leq n\) where equality implies a minimal immersion. The present paper uses techniques and has results similar to what can be found in \textit{B.-Y. Chen} [`Total mean curvature and submanifolds of finite type' (Ser. Pure Math. 1. Singapore: World Scientific) (1984; Zbl 0537.53049)] and \textit{B.-Y. Chen} and \textit{S. Jiang} [Bull. Belg. Math. Soc. -- Simon Stevin 2, No. 1, 75-85 (1995; Zbl 0827.53003)].