극소곡면

메모

• This was first considered by Lagrange (1762), who raised the question of existence of surfaces of least area having a given closed curve in three-space as the boundary. He derived the differential equation that must be satisfied by a function of two variables whose graph minimizes area among surfaces with a given contour.
• Later Meusnier discovered that this is equivalent to the vanishing of the mean curvature, and the study of the differential geometry of these surfaces was started.
• the plane in \(R^3\), helicoid, catenoid, Scherk surface, and Enneper surface
• representation formula of Enneper and Weierstrass which expresses a given minimal surface in terms of integrals involving a holomorphic function μ and a meromorphic function ν

관련도서

• Dierkes, Ulrich, Stefan Hildebrandt, Friedrich Sauvigny, Ruben Jakob, and Albrecht Küster. Minimal Surfaces. 2nd, rev. and enlarged ed. 2010 edition. Fort Worth: Springer, 2010.

리뷰, 에세이, 강의노트

• Harrison, Jenny, and Harrison Pugh. “Plateau’s Problem: What’s Next.” arXiv:1509.03797 [math], September 12, 2015. http://arxiv.org/abs/1509.03797.

관련논문

• William H. Meeks III, Joaquin Perez, Antonio Ros, Bounds on the topology and index of minimal surfaces, arXiv:1605.02501 [math.DG], May 09 2016, http://arxiv.org/abs/1605.02501
• Dey, Rukmini. “Ramanujan’s Identities, Minimal Surfaces and Solitons.” arXiv:1508.05183 [math], August 21, 2015. http://arxiv.org/abs/1508.05183.
• Chuaqui, Martin. “General Injectivity Criteria for Weierstrass-Enneper Lifts.” arXiv:1508.00432 [math], August 3, 2015. http://arxiv.org/abs/1508.00432.
• Jiang, YuePing, ZhiHong Liu, and Saminathan Ponnusamy. “Univalent Harmonic Mappings and Lift to the Minimal Surfaces.” arXiv:1508.00199 [math], August 2, 2015. http://arxiv.org/abs/1508.00199.
• Hoffman, David, Martin Traizet, and Brian White. “Helicoidal Minimal Surfaces of Prescribed Genus.” arXiv:1508.00064 [math], July 31, 2015. http://arxiv.org/abs/1508.00064.
• Marques, Fernando Coda. “Minimal Surfaces - Variational Theory and Applications.” arXiv:1409.7648 [math], September 26, 2014. http://arxiv.org/abs/1409.7648.
• Gutshabash, E. Sh. “Nonlinear Sigma Model, Zakharov-Shabat Method, and New Exact Forms of the Minimal Surfaces in \(R^3\).” arXiv:1409.6741 [math-Ph, Physics:nlin], September 23, 2014. http://arxiv.org/abs/1409.6741.