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Global existence of generalized rotational hypersurfaces with prescribed mean curvature in Euclidean spaces. I. (English) Zbl 1325.53017

Summary: We prove that for a given continuous function \(H(s)\), \((-\infty < s < \infty)\), there exists a globally defined generating curve of a rotational hypersurface in a Euclidean space such that the mean curvature is \(H(s)\). We also prove a similar theorem for generalized rotational hypersurfaces of \(O(l+1)\times O(m+1)\)-type. The key lemmas in this paper show the existence of solutions for singular initial value problems of ordinary differential equations satisfied using generating curves of those hypersurfaces.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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References:

[1] C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl. Ser.1, 6 (1841), 309-320.
[2] J. Dorfmeister and K. Kenmotsu, On a theorem by Hsiang and Yu, Ann. Global Anal. Geom., 33 (2008), 245-252. · Zbl 1151.53054
[3] J. Dorfmeister and K. Kenmotsu, Rotational hypersurfaces of periodic mean curvature, Differential Geom. Appl., 27 (2009), 702-712. · Zbl 1213.53076
[4] W.-Y. Hsiang, Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces, I, J. Differential Geom., 17 (1982), 337-356. · Zbl 0493.53043
[5] W.-Y. Hsiang and H.-L. Huynh, Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces, II, Pacific J. Math., 130 (1987), 75-95. · Zbl 0647.53042
[6] W.-Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geom., 5 (1971), 1-38. · Zbl 0219.53045
[7] W.-Y. Hsiang and W.-C. Yu, A generalization of a theorem of Delaunay, J. Differential Geom., 16 (1981), 161-177. · Zbl 0504.53044
[8] K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tôhoku Math. J. (2), 32 (1980), 147-153. · Zbl 0431.53005
[9] K. Kenmotsu, Surfaces of revolution with periodic mean curvature, Osaka J. Math., 40 (2003), 687-696. · Zbl 1041.53005
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