Download A survey on classical minimal surface theory by William H., III Meeks, Joaquin Perez PDF

By William H., III Meeks, Joaquin Perez

ISBN-10: 0821869124

ISBN-13: 9780821869123

Meeks and Pérez current a survey of contemporary impressive successes in classical minimum floor concept. The category of minimum planar domain names in third-dimensional Euclidean area offers the focal point of the account. The evidence of the type depends upon the paintings of many at the moment energetic prime mathematicians, hence making touch with a lot of crucial ends up in the sphere. throughout the telling of the tale of the category of minimum planar domain names, the overall mathematician may well trap a glimpse of the intrinsic great thing about this idea and the authors' standpoint of what's taking place at this historic second in a really classical topic. This e-book contains an up to date travel via a few of the contemporary advances within the thought, equivalent to Colding-Minicozzi idea, minimum laminations, the ordering theorem for the distance of ends, conformal constitution of minimum surfaces, minimum annular ends with limitless overall curvature, the embedded Calabi-Yau challenge, neighborhood images at the scale of curvature and topology, the neighborhood detachable singularity theorem, embedded minimum surfaces of finite genus, topological category of minimum surfaces, strong point of Scherk singly periodic minimum surfaces, and awesome difficulties and conjectures

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Singly-periodic Scherk surface with angle θ = π/2 (left), and its conjugate surface, the doubly-periodic Scherk surface (right). Images courtesy of M. Weber. Hoffman and White [92] gave a variational proof of the existence of a genusone helicoid. 21). The singly-periodic Scherk surfaces. 5 Left for the case θ = π/2. Discovered by Scherk [217] in 1835, these surfaces denoted by Sθ form a 1-parameter family of complete, embedded, genus-zero minimal surfaces in a quotient of R3 by a translation, and have four annular ends.

If e ∈ E(M ) is not a simple end (equivalently, if it is a limit point of E(M ) ⊂ [0, 1]), we will call it a limit end of M . When M has dimension 2, then an elementary topological analysis using compact exhaustions shows that an end e ∈ E(M ) is simple if and only if it 18 Throughout the paper, the word eventually for proper arcs means outside a compact subset of the parameter domain [0, ∞). 36 2 Basic results in classical minimal surface theory can be represented by a proper subdomain Ω ⊂ M with compact boundary which is homeomorphic to one of the following models: (a) S1 × [0, ∞) (this case is called an annular end).

5. Singly-periodic Scherk surface with angle θ = π/2 (left), and its conjugate surface, the doubly-periodic Scherk surface (right). Images courtesy of M. Weber. Hoffman and White [92] gave a variational proof of the existence of a genusone helicoid. 21). The singly-periodic Scherk surfaces. 5 Left for the case θ = π/2. Discovered by Scherk [217] in 1835, these surfaces denoted by Sθ form a 1-parameter family of complete, embedded, genus-zero minimal surfaces in a quotient of R3 by a translation, and have four annular ends.

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