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New examples of tubular minimal surfaces of arbitrary codimension. (English. Russian original) Zbl 0987.53004

Math. Notes 62, No. 1, 129-131 (1997); translation from Mat. Zametki 62, No. 1, 154-156 (1997).
From the text: Let \(F\) be a compact manifold, \((\alpha,\beta)\) an interval and let us consider a \(C^2\)-smooth function \(r(t)\) defined on \((\alpha,\beta)\). Then we define a \(p\)-dimensional surface \(M\) by the \(C^2\)-smooth embedding \(u: F\times (\alpha,\beta)\to \mathbb{R}^{n+1}\), \(u(y,t)= r(t)R(y)+ te_0\), where \(R: F\to S^{n-1}\subset\{0\}\times \mathbb{R}^n\subset \mathbb{R}^{n+1}\) is a \(C^2\)-smooth embedding of a \((p-1)\)-dimensional manifold \(F\).
Our main result is the following theorem. A surface \(M\) of the above form is a minimal tubular submanifold with projection \((\alpha,\beta)\) if and only if \(R: F\to S^{n-1}\) is a minimal embedding in the sphere and the function \(r(t)\) satisfies the differential equation \(r''(t)r(t)= (p-1)(1+ r^{\prime 2}(t))\).

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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