Gap theorems for minimal submanifolds in \(\mathbb{R}^{n+1}\)
Let \(M^n\subset \mathbb{R}^N\) a complete minimal immersed submanifold of \(\mathbb{R}^N\), \(n<N\), and let \(|A|\) be the norm of the second fundamental form of \(M^n\). The following two gap theorems are proved in this paper. Theorem A. Let \(N=n+1\), \(n\geq 4\). There exists a constant \(C_1(n)\), depending only on \(n\), such that if \[ \left( \int_M|A|^n dM^n \right)^{1/n} < C_1, \] then \(M\) is a hyperplane. Theorem B. Let \(N\) be arbitrary and \(n\geq 3\). There exists a constant \(C_2(n)\), depending only on \(n\), such that if \[ \left( \int_M|A|^n dM^n \right)^{1/n} < C_2, \] then \(M\) has only one end. Theorem B is used to prove Theorem A. The constants \(C_1\) and \(C_2\) depend on appropriate Sobolev inequalities. It is not known whether Theorem A holds for arbitrary \(N\).
scientific article; zbMATH DE number 1833135