Rigidity of minimal submanifolds with flat normal bundle
Let \(M^n\) be an \(n\)-dimensional complete minimal immersed submanifold in \(\mathbb{R}^{n+p}\). For some number \(0<\delta \leq 1\), we say that \(M\) is \(\delta\)-super-stable if \(0 \leq \int_M (|\nabla f|^2 -\delta |A|^2 f^2 )\), for any \(f\in C_{0}^\infty (M)\), where \(|A|\) denotes the norm of the second fundamental form of \(M\). Assume that \(M^n\), \(n\geq 3\), is a \(\delta\)-super-stable complete immersed minimal submanifold in \(\mathbb{R}^{n+p}\) with flat normal bundle. The main results of this paper are the following (Theorem 1.1) : {\parindent5mm \begin{itemize}\item[1.] If \(\delta >\frac{n-2}{n}\) and \(\lim_{R\rightarrow \infty} \frac{1}{R^{2+2q\delta} }\int_{B(2R)\setminus B(R)}|A|^{2\delta}=0\), where \(q<\sqrt{(\frac{2}{n}-1){\delta}^{-1} +1}\), then \(M\) is an affine plane. \item[2.] If \(\delta >\frac{n-2}{n}\) and \(\lim_{R\rightarrow \infty} \frac{1}{R^2}\int_{B(2R)\setminus B(R)}|A|^{\frac{2(n-2)}{n}}=0\), then \(M\) is an affine plane. \item[3.] If \(\delta = \frac{n-2}{n}\) and \(\lim_{R\rightarrow \infty} \frac{1}{R^2}\int_{B(2R)\setminus B(R)}|A|^{\frac{2(n-2)}{n}}=0\), then \(M\) is a catenoid in some Euclidean subspace or an affine plane. \end{itemize}} These results extend the results of \textit{K. Seo} [Commun. Korean Math. Soc. 23, No.~3, 421--426 (2008; Zbl 1168.53323)] and \textit{L. F. Tam} and \textit{D. T. Zhou} [Proc. Am. Math. Soc. 137, No.~10, 3451--3461 (2009; Zbl 1184.53016)].
scientific article; zbMATH DE number 5826108
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