Complete minimal surfaces in R.
Francisco J. López, Francisco Martín (1999)
Publicacions Matemàtiques
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In this paper we review some topics on the theory of complete minimal surfaces in three dimensional Euclidean space.
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Complete minimal surfaces in R.
Curvature estimates for minimal surfaces in -manifolds
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Existence results for embedded minimal surfaces of controlled topological type, I
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A rigidity theorem for Riemann's minimal surfaces
Pascal Romon (1993)
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We describe first the analytic structure of Riemann’s examples of singly-periodic minimal surfaces; we also characterize them as extensions of minimal annuli bounded by parallel straight lines between parallel planes. We then prove their uniqueness as solutions of the perturbed problem of a punctured annulus, and we present standard methods for determining finite total curvature periodic minimal surfaces and solving the period problems.
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Reinhold Böhme (1981-1982)
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Invariants and Bonnet-type theorem for surfaces in ℝ4
Georgi Ganchev, Velichka Milousheva (2010)
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In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes...
Linearization and explicit solutions of the minimal surface equations.
Alexander G. Reznikov (1992)
Publicacions Matemàtiques
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We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Δh + 2h = 0 describing locally germs of minimal surfaces. Here Δ is the Laplace-Beltrami operator on the standard two-dimensional sphere. It explains the existence of the sum operator of minimal surfaces, introduced recently. In 4-dimensional space the equation Δ h + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero.