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

By William H., III Meeks, Joaquin Perez

Meeks and Pérez current a survey of modern mind-blowing successes in classical minimum floor idea. The class of minimum planar domain names in third-dimensional Euclidean house presents the focal point of the account. The evidence of the category relies on the paintings of many presently energetic prime mathematicians, therefore making touch with a lot of crucial ends up in the sector. during the telling of the tale of the category of minimum planar domain names, the final mathematician may possibly capture a glimpse of the intrinsic great thing about this concept and the authors' viewpoint of what's taking place at this historic second in a truly classical topic. This ebook contains an up-to-date journey via many of the contemporary advances within the idea, comparable to Colding-Minicozzi concept, minimum laminations, the ordering theorem for the gap of ends, conformal constitution of minimum surfaces, minimum annular ends with countless overall curvature, the embedded Calabi-Yau challenge, neighborhood photos 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 extraordinary difficulties and conjectures

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4. 4 insures that any non-planar, properly embedded, one-ended, minimal surface with finite topology must be necessarily asymptotic to a helicoid with finitely many handles, and it can be described analytically by meromorphic data (dg/g, dh) on a compact Riemann surface by means of the classical Weierstrass representation. Regarding one-ended surfaces with infinite topology, Callahan, Hoffman and Meeks [17] showed that any non-flat, doubly or triply-periodic minimal surface in R3 must have infinite genus and only one end.

2. Any isolated point e ∈ E(M ) is called a simple end of M . 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).

1. Left: The catenoid. Center: The helicoid. Right: The Enneper surface. Images courtesy of M. Weber. 5. Some interesting examples of complete minimal surfaces We will now use the Weierstrass representation for introducing some of the most celebrated complete minimal surfaces. 5. The plane. M = C, g(z) = 1, dh = dz. It is the only complete, flat minimal surface in R3 . The catenoid. 1 Left. In 1741, Euler [56] discovered that when a catenary x1 = cosh x3 is rotated around the x3 -axis, one obtains a surface which minimizes area among surfaces of revolution after prescribing boundary values for the generating curves.

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