Delaunay surfaces expressed in terms of a Cartan moving frame

• Autorzy: Bracken Paul

Identyfikatory

DOI

10.1515/jaa-2020-2012

• Języki publikacji: EN

Abstrakty

EN

Delaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

Słowa kluczowe

EN

metric; geometry; manifold; fundamental form; structure equation; Delaunay

PL

miara; geometria; rozmaitość; forma podstawowa; równania struktury; Delaunay

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2020
• Tom: Vol. 26, nr 1
• Strony: 153--160
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Bracken Paul

• Department of Mathematics, University of Texas, Edinburg, TX 78540, USA

Bibliografia

• [1] P. Bracken and A. M. Grundland, On certain classes of solutions of the Weierstrass-Enneper system inducing constant mean curvature surfaces, J. Nonlinear Math. Phys. 6 (1999), no. 3, 294-313.
• [2] P. Bracken and A. M. Grundland, Symmetry properties and explicit solutions of the generalized Weierstrass system, J. Math. Phys. 42 (2001), no. 3, 1250-1282.
• [3] S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, World Scientific, Singapore, 1999.
• [4] C. Delaunay, Sur la surface de revolution dont la courbure est constante, J. Math. Pures Appl. 6 (1841), 309-320.
• [5] A. G. Greenhill, The Applications of Elliptic Functions, Dover, New York, 1959.
• [6] J. Hass and R. Schlafly, Double bubbles minimize, Ann. of Math. (2) 151 (2000), no. 2, 459-515.
• [7] K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tohoku Math. J. (2) 32 (1980), no. 1, 147-153.
• [8] K. Kenmotsu, Surfaces with Constant Mean Curvature, Transl. Math. Monogr. 221, American Mathematical Society, Providence, 2003.
• [9] B. G. Konopelchenko and I. A. Taimanov, Constant mean curvature surfaces via an integrable dynamical system, J. Phys. A 29 (1996), no. 6, 1261-1265.
• [10] A. Korn, Zwei Anwendungen der Methode der sukzessiven Annäherungen, Schwarz-Festschr. (1916), 215-219.
• [11] J. Lichtenstein, Zur Theorie der konformen Abbildung, Bull. Internat. Acad. Sci. Crecivie CI. Sci. Math. Nat. Ser. A 1916 (1916), 192-217.
• [12] I. M. Mladenov, Conformal immersions of Delaunay surfaces and their duals, in: Geometry, Integrability and Quantization, Softex, Sofia (2004), 158-168.
• [13] N. Sultana, Explicit parametrization of Delaunay surfaces in space forms via loop group methods, Kobe J. Math. 22 (2005), no. 1-2, 71-107.
• [14] T. J. Willmore, An Introduction to Differential Geometry, 2nd ed., Oxford University, Oxford, 1959.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).