By William H., III Meeks, Joaquin Perez

Meeks and Pérez current a survey of contemporary fantastic successes in classical minimum floor idea. The type of minimum planar domain names in third-dimensional Euclidean area presents the focal point of the account. The evidence of the class will depend on the paintings of many at the moment lively top mathematicians, hence making touch with a lot of an important ends up in the sphere. in the course of the telling of the tale of the type of minimum planar domain names, the final mathematician may perhaps seize a glimpse of the intrinsic great thing about this concept and the authors' point of view of what's taking place at this historic second in a really classical topic. This e-book contains an up-to-date journey via many of the contemporary advances within the thought, resembling Colding-Minicozzi idea, 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 photographs at the scale of curvature and topology, the neighborhood detachable singularity theorem, embedded minimum surfaces of finite genus, topological type of minimum surfaces, area of expertise of Scherk singly periodic minimum surfaces, and amazing difficulties and conjectures

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**Extra resources for A survey on classical minimal surface theory**

**Sample text**

The second statement in the above theorem was motivated by a result of Meeks [128], who proved that every properly embedded minimal surface in T2 × R has a ﬁnite number of ends; hence, in this setting ﬁnite genus implies ﬁnite total curvature. Meeks and Rosenberg [153, 156] also studied the asymptotic behavior of complete, embedded minimal surfaces M with ﬁnite total curvature in R3 /G. 5; such ends are called planar ends), to ends of quotient helicoids (called helicoidal ends), or to ﬂat annuli (as in the singly or doubly-periodic Scherk minimal surfaces; for this reason, such ends are called Scherk-type ends).

Meeks and Yau proved that the metric g on W can be approximated by a family of homogeneously regular metrics {gn }n∈N on W , which converges smoothly on compact subsets of W to g, and each gn satisﬁes a convexity condition outside of W ⊂ W , which forces the leastarea surface ΣΓ (gn ) to lie in W if Γ lies in W . A subsequence of the ΣΓ (gn ) converges to a smooth minimal surface ΣΓ of least-area in W with respect to the original ﬂat metric, thereby ﬁnishing our description of the barrier construction.

When a = 0, the end is called a catenoidal end; if a = 0, we have a planar end. 5 for a description of the catenoid) and a planar end is asymptotic to the end of a plane. In particular, complete embedded minimal surfaces with ﬁnite total curvature are always proper; in fact, an elementary analysis of the asymptotic behavior shows that the equivalence between completeness and properness still holds for immersed minimal surfaces with ﬁnite total curvature. As explained above, minimal surfaces with ﬁnite total curvature have been widely studied and their comprehension is the starting point for dealing with more general questions about complete embedded minimal surfaces.