A gap theorem for minimal submanifolds in Euclidean space. (Un théorème de seuil pour les sous-variétés minimales dans l’espace euclidien.) (English. French summary) Zbl 1308.53022
Summary: We prove that for a complete minimal submanifold \(M^n\) immersed in the Euclidean space \(\mathbb R^{n+d}\), if the second fundamental form \(A\) and the intrinsic distance function \(r\) from a fixed point satisfy \(r(x)| A| (x)\leq\varepsilon\) for all \(x\in M\), where \(\epsilon\) is a positive constant depending only on \(n\), then \(M\) is an affine subspace of \(\mathbb R^{n+d}\).
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
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