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

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

Show description

Read Online or Download A survey on classical minimal surface theory PDF

Best differential geometry books

Geodesic and Horocyclic Trajectories (Universitext)

In past times thirty years, powerful relationships have interwoven the fields of dynamical structures, linear algebra and quantity concept. This rapport among varied components of arithmetic has enabled the solution of a few very important conjectures and has in reality given start to new ones. This ebook sheds mild on those relationships and their purposes in an trouble-free environment, by way of exhibiting that the research of curves on a floor may end up in orbits of a linear crew or perhaps to persevered fraction expansions of genuine numbers.

Least action principle of crystal formation of dense packing type and Kepler's conjecture

The dense packing of microscopic spheres (i. e. atoms) is the elemental geometric association in crystals of mono-atomic parts with vulnerable covalent bonds, which achieves the optimum "known density" of B/ 18. In 1611, Johannes Kepler had already "conjectured" that B/ 18 might be the optimum "density" of sphere packings.

Basic Concepts of Synthetic Differential Geometry

Beginning at an introductory point, the e-book leads quickly to special and infrequently new leads to artificial differential geometry. From rudimentary research the ebook strikes to such vital effects as: a brand new facts of De Rham's theorem; the bogus view of worldwide motion, going so far as the Weil attribute homomorphism; the systematic account of dependent Lie gadgets, similar to Riemannian, symplectic, or Poisson Lie gadgets; the view of worldwide Lie algebras as Lie algebras of a Lie team within the artificial feel; and finally the artificial building of symplectic constitution at the cotangent package deal as a rule.

Coulomb Frames in the Normal Bundle of Surfaces in Euclidean Spaces: Topics from Differential Geometry and Geometric Analysis of Surfaces

This e-book is meant for complicated scholars and younger researchers attracted to the research of partial differential equations and differential geometry. It discusses ordinary ideas of floor geometry in higher-dimensional Euclidean areas, specifically the differential equations of Gauss-Weingarten including numerous integrability stipulations and corresponding floor curvatures.

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.

Download PDF sample

Rated 4.01 of 5 – based on 16 votes