Abstract
The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss–Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with vanishing Gauss–Kronecker curvature and scalar curvature bounded from below.
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Asperti, A.C., Chaves, R.M.B., Sousa, L.A.M.: The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms. Math. Z. 267, 523–533 (2010)
Beez, R.: Zur Theorie des Krümmungsmasses von Mannigfaltigkeiten höhere Ordnung. Z. Angew. Math. Phys. 21, 373–401 (1876)
Bryant, R.L.: Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Differ. Geom. 17, 455–473 (1982)
Bryant, R.L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 23–53 (1984)
Cartan, E.: La déformation des hypersurfaces dans l’espace euclidien réel a n dimensions. Bull. Soc. Math. Fr. 44, 65–99 (1916)
de Almeida, S.C., Brito, F.G.B.: Minimal hypersurfaces of \(\mathbb{S}^{4}\) with constant Gauss–Kronecker curvature. Math. Z. 195, 99–107 (1987)
Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
Eschenburg, J.-H., Tribuzy, R.: Branch points of conformal mappings of surfaces. Math. Ann. 279, 621–633 (1988)
Evans, L., Gariepy, R.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Dajczer, M., Gromoll, D.: Gauss parametrizations and rigidity aspects of submanifolds. J. Differ. Geom. 22, 1–12 (1985)
Ferus, D.: On the type number of hypersurfaces in spaces of constant curvature. Math. Ann. 187, 310–316 (1970)
Gulliver II, R.D., Osserman, R., Royden, H.L.: A theory of branched immersions of surfaces. Am. J. Math. 95, 750–812 (1973)
Hasanis, T., Savas-Halilaj, A., Vlachos, T.: Minimal hypersurfaces with zero Gauss–Kronecker curvature. Ill. J. Math. 49, 523–529 (2005)
Hasanis, T., Savas-Halilaj, A., Vlachos, T.: Complete minimal hypersurfaces of \(\mathbb{S}^{4}\) with zero Gauss–Kronecker curvature. Math. Proc. Camb. Philos. Soc. 142, 125–132 (2007)
Hasanis, T., Savas-Halilaj, A., Vlachos, T.: Complete minimal hypersurfaces in the hyperbolic space ℍ4 with vanishing Gauss–Kronecker curvature. Trans. Am. Math. Soc. 359, 2799–2818 (2007)
Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions, 2nd edn. Birkhäuser, Boston (2002)
Killing, W.: Die nicht-Euklidischen Raumformen in Analytische Behandlung. Teubner, Leipzig (1885)
Lawson, H.B.: Complete minimal surfaces in \(\mathbb{S}^{3}\). Ann. Math. 92, 335–374 (1970)
Meier, M.: Removable singularities of harmonic maps and an application to minimal submanifolds. Indiana Univ. Math. J. 35, 705–726 (1986)
Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)
Palais, R.: A global formulation of the Lie theory of transformation groups. Mem. Amer. Math. Soc. 22 (1957)
Ramanathan, J.: Minimal hypersurfaces in \(\mathbb{S}^{4}\) with vanishing Gauss–Kronecker curvature. Math. Z. 205, 645–658 (1990)
Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. Éc. Norm. Super. 11, 211–228 (1978)
Sbrana, V.: Sulla varieta ad (n−1)-dimensione deformabili nello spazio euclideo ad n-dimensione. Rend. Circ. Mat. Palermo 27, 1–45 (1909)
Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
Wood, C.M.: The energy of a unit vector field. Geom. Dedic. 64, 319–330 (1997)
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Communicated by Alexander Isaev.
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Savas-Halilaj, A. On Deformable Minimal Hypersurfaces in Space Forms. J Geom Anal 23, 1032–1057 (2013). https://doi.org/10.1007/s12220-011-9272-2
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DOI: https://doi.org/10.1007/s12220-011-9272-2