The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

Meeks, William H.; Pérez, Joaquín; Ros, Antonio

doi:10.1007/s00222-004-0374-3

Cited by 27 publications

(50 citation statements)

References 37 publications

(57 reference statements)

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“…Since outside a compact subset of M the surface has genus zero, the blown-up limit M ∞ has genus zero. By the classification results in [16] for such M ∞ , on this scale, the surface M has the appearance of either a catenoid or a two limit end genus zero minimal surface near p n .…”

Section: Length(γmentioning

confidence: 90%

The minimal lamination closure theorem

2006

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We prove that the closure of a complete embedded minimal surface M in a Riemannian three-manifold N has the structure of a minimal lamination, when M has positive injectivity radius. When N is R 3 , we prove that such a surface M is properly embedded. Since a complete embedded minimal surface of finite topology in R 3 has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi: A complete embedded minimal surface of finite topology in R 3 is proper. More generally, we prove that if M is a complete embedded minimal surface of finite topology and N has nonpositive sectional curvature (or is the Riemannian product of a homogeneously regular Riemannian surface with R), then the closure of M has the structure of a minimal lamination. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

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Section: Length(γmentioning

confidence: 90%

The minimal lamination closure theorem

2006

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“…We call this procedure blowing-up by the scale of topology. This scale was defined and used in [74,75] to prove that any properly embedded minimal surface of finite genus has bounded curvature and is recurrent for Brownian motion. We now explain the elements of this new scale.…”

Section: Sup |K Mn∩b(s(t)r) | → ∞ As N → ∞ For Any T ∈ R and R >mentioning

confidence: 99%

“…Using the uniformly locally simply connected property of {M n } n , we prove that its limits are properly embedded nonsimply connected minimal surfaces with genus at most g and possibly infinitely many ends. The infinite topology limits are discarded by an application of either a descriptive Theorem of the geometry of properly embedded minimal surfaces in R 3 with finite genus and two limit ends (Meeks, Pérez and Ros [74], see also Theorem 32 below), or a nonexistence Theorem for properly embedded minimal surfaces in R 3 with finite genus and one limit end (Meeks, Pérez and Ros [75] or Theorem 31 below). Hence, any possible limit M of a subsequence of {M n } n must be a finite total curvature surface or a helicoid with positive genus at most g. A surgery argument allows one to modify the surfaces M n by replacing compact pieces of M n close to the limit M by a finite number of disks, obtaining a new surface M n with strictly less topology than M n and which is not minimal in the replaced part.…”

Section: Theorem 29 [76] For Every Nonnegative Integer G There Exismentioning

confidence: 99%

“…A crucial result by Collin, Kusner, Meeks and Rosenberg [26] (Theorem 19) insures that M has no middle limit ends, hence either it has one limit end (this one being the top or the bottom limit end) or both top and bottom ends are the limit ends of M , like in a Riemann minimal example. Very recently, Meeks, Pérez and Ros [75] have discarded the one limit end case through the following result. Sketch of the proof.…”

Section: Corollary 3 For Every Nonnegative Integer G There Exists Amentioning

confidence: 99%

“…An important uniqueness for certain parking garage structures was recently given by Weber and Wolf [126], who proved that the columns of these parking garages project to the zeros of Hermite polynomials on the real line; this uniqueness result plays a crucial role in their proposed existence proof of an embedded genus g helicoid for every positive integer g. In Section 10 we cover some topological aspects of properly embedded minimal surfaces. These results include a discussion of the recent Topological Classification Theorem for Minimal Surfaces by Frohman and Meeks [38] and the recent topological obstructions of Meeks, Pérez and Ros [75,76] for properly embedded minimal surfaces of finite genus. A particular consequence of these new topological obstructions is that every properly embedded minimal surface of finite genus in R 3 is recurrent for Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

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Conformal properties in classical minimal surface theory

Meeks

Pérez

rez

2004

Surveys in Differential Geometry

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Abstract. This is a survey of recent developments in the classical theory of minimal surfaces in R 3 with an emphasis on the conformal properties of these surfaces such as recurrence and parabolicity. We cover the maximum principle at infinity for properly immersed minimal surfaces in R 3 and some new results on harmonic functions as they relate to the classical theory. We define and demonstrate the usefulness of universal superharmonic functions. We present the compactness and regularity theory of Colding and Minicozzi for limits of sequences of simply connected minimal surfaces and its application by Meeks and Rosenberg in their proof of the uniqueness of the helicoid. Finally, we discuss some recent deep results on the topology and on the index of the stability operator of properly embedded minimal surfaces and give an application of the classical Shiffman Jacobi function to the classification of minimal surfaces of genus zero.

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Bending the helicoid

Meeks

Weber

2007

Math. Ann.

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We construct Colding-Minicozzi limit minimal laminations in open domains in R 3 with the singular set of C 1 -convergence being any properly embedded C 1,1 -curve. By Meeks' C 1,1 -regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination L is a locally finite collection S(L) of C 1,1 -curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding-Minicozzi limit minimal lamination. In the case the curve is the unit circle S 1 (1) in the (x 1 , x 2 )-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H n of finite total curvature which contain the circle. The complete minimal surfaces H n contain embedded compact minimal annuli H n in closed compact neighborhoods N n of the circle that converge as n → ∞ to R 3 − x 3 -axis. In this case, we prove that the H n converge on compact sets to the foliation of R 3 − x 3 -axis by vertical half planes with boundary the x 3 -axis and with S 1 (1) as the singular set of C 1 -convergence. The H n have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.

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The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

Cited by 27 publications

References 37 publications

The minimal lamination closure theorem

The minimal lamination closure theorem

Conformal properties in classical minimal surface theory

Bending the helicoid

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