Abstract
Minimal surfaces are classically defined as surfaces of zero mean curvature in ℝ3. Choosing conformal parameters, one may extend this definition by requiring that a minimal surface X:Ω→ℝ3 is a nonconstant harmonic mapping in conformal representation, and similarly surfaces of prescribed mean curvature are defined. This allows for isolated singular points of X in Ω, so-called branch points, which are discussed in later chapters. If a minimal surface can be represented as graph of a scalar function, one is led to a nonparametric minimal surface satisfying the minimal surface equation. For entire solutions of this equation the celebrated Bernstein theorem is derived as well as a curvature estimate due to E. Heinz which in turn implies Bernstein’s result.
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© 2010 Springer-Verlag Berlin Heidelberg
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Dierkes, U., Hildebrandt, S., Sauvigny, F. (2010). Minimal Surfaces. In: Minimal Surfaces. Grundlehren der mathematischen Wissenschaften, vol 339. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11698-8_2
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DOI: https://doi.org/10.1007/978-3-642-11698-8_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11697-1
Online ISBN: 978-3-642-11698-8
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