By William H., III Meeks, Joaquin Perez

Meeks and Pérez current a survey of contemporary outstanding successes in classical minimum floor idea. The class of minimum planar domain names in 3-dimensional Euclidean area offers the focal point of the account. The evidence of the class relies on the paintings of many presently lively top mathematicians, therefore making touch with a lot of an important leads to the sphere. during the telling of the tale of the type of minimum planar domain names, the overall mathematician may possibly trap a glimpse of the intrinsic great thing about this thought and the authors' point of view of what's occurring at this historic second in a truly classical topic. This booklet comprises an up-to-date travel via the various fresh advances within the conception, comparable 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 images 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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**Example text**

2 (Monotonicity Formula [27, 107]). Let X : M → R3 be a connected, properly immersed minimal surface. Given p ∈ R3 , let A(R) be the area of the portion of X(M ) inside a ball of radius R > 0 centered at p. Then, A(R)R−2 is non-decreasing. In particular, limR→∞ A(R)R−2 ≥ π with equality if and only if M is a plane. 1) is that they satisfy certain maximum principles. We will state them for minimal surfaces in R3 , but they also hold when the ambient space is any Riemannian threemanifold. 3 (Interior Maximum Principle [220]).

3), the area functional A = A(t) for this deformation has A (0) = 0. 12) Ω where K is the Gaussian curvature function of M and Δ its Laplace operator. 12) can be viewed as the bilinear form associated to the linear elliptic L2 -selfadjoint operator L = Δ − 2K = Δ + |∇N |2 , which is usually called the Jacobi operator or stability operator. 1. A C 2 -function u : M → R satisfying Δu − 2Ku = 0 on M is called a Jacobi function. We will let J (M ) denote the linear space of Jacobi functions on M . Classical elliptic theory implies that given a subdomain Ω ⊂ M with compact closure, the Dirichlet problem for the Jacobi operator in Ω has an inﬁnite discrete spectrum {λk }k∈N∪{0} of eigenvalues with λk +∞ as k goes to inﬁnity, and each eigenspace is a ﬁnite dimensional linear subspace of C ∞ (Ω) ∩ H01 (Ω), where H01 (Ω) denotes the usual Sobolev space of L2 -functions with L2 weak partial derivatives and trace zero.

Half-planes with boundary the x3 -axis, and with S1 as the singular set of C 1 -convergence. The method applied by Meeks, Weber and Mira to ﬁnd the bent helicoids is the classical Bj¨ orling formula [186] with an orthogonal unit ﬁeld along S1 that spins an arbitrary number n of times around the circle. This construction also makes sense when n is half an integer; in the case n = 12 , H1/2 is the double cover of the Meeks minimal M¨ obius strip described in the previous example. 3 below) for the singular set of convergence in a Colding-Minicozzi limit minimal lamination.