By William H., III Meeks, Joaquin Perez

Meeks and Pérez current a survey of contemporary unbelievable successes in classical minimum floor thought. The type of minimum planar domain names in 3-dimensional Euclidean area offers the point of interest of the account. The facts of the class relies on the paintings of many at present energetic best mathematicians, hence making touch with a lot of crucial ends up in the sphere. during the telling of the tale of the type of minimum planar domain names, the final mathematician may possibly trap a glimpse of the intrinsic great thing about this conception and the authors' point of view of what's taking place at this old second in a truly classical topic. This ebook comprises an up to date travel via the various contemporary advances within the thought, equivalent to Colding-Minicozzi thought, 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 photos at the scale of curvature and topology, the neighborhood detachable singularity theorem, embedded minimum surfaces of finite genus, topological category of minimum surfaces, forte of Scherk singly periodic minimum surfaces, and notable difficulties and conjectures

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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. This surface was called the alysseid or since Plateau’s time, the catenoid. 1). This surface has genus zero, two ends and total curvature −4π. Together with the plane, the catenoid is the only minimal surface of revolution (Bonnet [11]) and the unique complete, embedded minimal surface with genus zero, ﬁnite topology and more than one end (L´ opez and Ros [117]).

The Meeks minimal M¨ obius strip. 2 Left. Found by Meeks [124], the minimal surface deﬁned by this Weierstrass data double covers a complete, immersed minimal surface M1 ⊂ R3 which is topologically a M¨ obius strip. This is the unique complete, minimally immersed surface in R3 of ﬁnite total curvature −6π. It contains a unique closed geodesic which is a planar circle, and also contains a line bisecting the circle. n n −n z +z The bent helicoids. 2 Right. Discovered by Meeks and Weber [162] and independently by Mira [172], these are complete, immersed minimal annuli Hn ⊂ R3 with two non-embedded ends and ﬁnite total curvature; each of the surfaces Hn contains the unit circle S1 in the (x1 , x2 )-plane, and a neighborhood of S1 in Hn contains an embedded annulus Hn which approximates, for n large, a highly spinning helicoid whose usual straight axis has been periodically bent into the unit circle S1 (thus the name of bent helicoids).

3. Left: The Costa torus. Center: A CostaHoﬀman-Meeks surface of genus 20. Right: Deformed Costa. Images courtesy of M. Weber. Costa [43, 44]. This is a thrice punctured torus with total curvature −12π, two catenoidal ends and one planar middle end. Costa [44] demonstrated the existence of this surface but only proved its embeddedness outside a ball in R3 . Hoﬀman and Meeks [84] demonstrated its global embeddedness, thereby disproving a longstanding conjecture that the only complete, embedded minimal surfaces in R3 of ﬁnite topological type are the plane, catenoid and helicoid.