Ando, N. Surfaces with zero mean curvature vector in neutral 4-manifolds. (English) Zbl 1445.53011 Differ. Geom. Appl. 72, Article ID 101647, 30 p. (2020). Summary: Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and para-Kähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces. Cited in 6 Documents MSC: 53B25 Local submanifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:zero mean curvature vector; neutral manifold; twistor space; holomorphic quartic differential; conformal Gauss map; Willmore type PDFBibTeX XMLCite \textit{N. Ando}, Differ. Geom. Appl. 72, Article ID 101647, 30 p. (2020; Zbl 1445.53011) Full Text: DOI References: [1] Ando, N., Local characterizations of complex curves in \(\boldsymbol{C}^2\) and sphere Schwarz maps, Int. J. Math., 27, Article 1650067 pp. (2016) · Zbl 1345.53061 [2] Ando, N., Complex curves and isotropic minimal surfaces in hyperKähler 4-manifolds, (Recent Topics in Differential Geometry and Its Related Fields (2019), World Scientific), 45-61 [3] Ando, N., Surfaces in pseudo-Riemannian space forms with zero mean curvature vector, Kodai Math. J., 43, 193-219 (2020) · Zbl 1479.53021 [4] N. Ando, K. Hamada, K. Hashimoto, S. Kato, Regularity of ends of zero mean curvature surfaces in \(\mathbf{R}^{2 , 1} \), in preparation. [5] Bryant, R., Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differ. Geom., 17, 455-473 (1982) · Zbl 0498.53046 [6] Bryant, R., A duality theorem for Willmore surfaces, J. Differ. Geom., 20, 23-53 (1984) · Zbl 0555.53002 [7] Davidov, J.; Grantcharov, G.; Mushkarov, O.; Yotov, M., Compact complex surfaces with geometric structures related to split quaternions, Nucl. Phys. B, 865, 330-352 (2012) · Zbl 1262.81130 [8] Eells, J.; Salamon, S., Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 12, 589-640 (1985) · Zbl 0627.58019 [9] Friedrich, T., On surfaces in four-spaces, Ann. Glob. Anal. Geom., 2, 257-287 (1984) · Zbl 0562.53039 [10] Hasegawa, K.; Miura, K., Extremal Lorentzian surfaces with null τ-planar geodesics in space forms, Tohoku Math. J., 67, 611-634 (2015) · Zbl 1334.53057 [11] Jensen, G.; Rigoli, M., Neutral surfaces in neutral four-spaces, Matematiche, 45, 407-443 (1990) · Zbl 0757.53035 [12] Kamada, H., Neutral hyperKähler structures on primary Kodaira surfaces, Tsukuba J. Math., 23, 321-332 (1999) · Zbl 0948.53023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.