We also obtain some estimates of the Gaussian curvature of the minimal surfaces in 3-dimensional Euclidean space R3 and of the spacelike minimal surfaces in L3.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:We prove a growth theorem for a function to belong to the class \sum(m;a) and generalize a Weierstrass-Enneper representation type theorem for the minimal surfaces given in [5] to spacelike minimal surfaces which lie in 3-dimensional Lorentz-Minkowski space L3.
📊
Key Findings & Data:We prove a growth theorem for a function to belong to the class \sum(m;a) and generalize a Weierstrass-Enneper representation type theorem for the minimal surfaces given in [5] to spacelike minimal surfaces which lie in 3-dimensional Lorentz-Minkowski space L3.
💡
Conclusion & Impact:We also obtain some estimates of the Gaussian curvature of the minimal surfaces in 3-dimensional Euclidean space R3 and of the spacelike minimal surfaces in L3.
We prove a growth theorem for a function to belong to the class \sum(m;a) and generalize a Weierstrass-Enneper representation type theorem for the minimal surfaces given in [5] to spacelike minimal surfaces which lie in 3-dimensional Lorentz-Minkowski space L3. We also obtain some estimates of the Gaussian curvature of the minimal surfaces in 3-dimensional Euclidean space R3 and of the spacelike minimal surfaces in L3.
Özet
We prove a growth theorem for a function to belong to the class \sum(m;a) and generalize a Weierstrass-Enneper representation type theorem for the minimal surfaces given in [5] to spacelike minimal surfaces which lie in 3-dimensional Lorentz-Minkowski space L3. We also obtain some estimates of the Gaussian curvature of the minimal surfaces in 3-dimensional Euclidean space R3 and of the spacelike minimal surfaces in L3.
TAŞTAN, H. M., POLATOĞLU, Y. (2022). On quasiconformal harmonic mappings lifting to minimal surfaces. Turkish Journal of Mathematics, Cilt: 37 - Sayı: 2, 267–277. https://ajindex.com/articles/608355
Ahlfors, L.: Lectures on quasiconformal mappings, Van Nostrand Princeton, N.J., 1966.
Chuaqui, M., Duren, P. and Osgood, B.: The Schwarzian derivative for harmonic mappings, J. Analyse Math. 91, 329–351 (2003).
Dey, R.: The Weierstrass-Enneper representation using hodographic coordinates on a minimal surface, Proc. Indian Acad. Sci.(Math.Sci) 113, No. 2, 189–193 (2003).
Duren, P.: Univalent functions, Grundlehren der Mathematiscen Wissenchaften 259, Springer-Verlag, Berlin, New York, 1983.
Duren, P.: Harmonic mappings in the plane, Cambridge University Press, Cambridge U.K, 2004. Ferr´ andez, A. and Lucas, P.: On surfaces in the 3-dimensional Lorentz-Minkowski space, Pacific J. Math. 152, No. 1, 93–100 (1992).
Kobayashi, O.: Maximal surfaces in the 3-dimensional Minkowski space Ä 3, Tokyo J. Math. 6, No. 2, 297–309 (1983).
Lehto, O., Virtanen, K. I.: Quasiconformal mappings in the plane, Second edition. Translated from the German by Lucas, K.W. Die Grundlehren der mathematischen Wissenhaften, Band 126. Springer-Verlag, New York-Heidelberg, 19 Lewandoski, Z.: Starlike madorants and subordination, Annales Universitatis, Mariae-Curie Sklodowska, LublinPolonia, XV. Sectio A, 79–84 (1961).
Lira, J. H., Melo, M. and Mercuri, M.: A Weierstrass representation for minimal surfaces in 3-dimensional manifolds, Results. Math. 60, 311–323 (2011).
Martio, O.: On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 425, 3–10 (1968). Ta¸stan, H. M. and Polato˘ glu, Y.: Growth properties for Weierstrass-Enneper representation of a minimal surface, Acta Universitatis Apulensis 20, 55–62 (2009).