General curvature estimates for stable H-surfaces in 3-manifolds applications

Rosenberg, Harold; Souam, Rabah; Toubiana, Éric

doi:10.4310/jdg/1279114303

Cited by 79 publications

(107 citation statements)

References 15 publications

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“…We remark that Rosenberg, Souam and Toubiana [183] have obtained a version of Theorem 2.27 valid in the two-sided case when the ambient three-manifold has a bound on its sectional curvature.…”

Section: Definition 221 Any Isolated Point E ∈ E(m ) Is Called a Simentioning

confidence: 99%

The classical theory of minimal surfaces

Meeks¹,

Pérez²

2011

Bull. Amer. Math. Soc.

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Abstract. We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid and the one-parameter family {R t } t∈(0,1) of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in R 3 ; the proof of this result depends primarily on work of Colding and Minicozzi, Collin, López and Ros, Meeks, Pérez and Ros, and Meeks and Rosenberg. Rather than culminating and ending the theory with this classification result, significant advances continue to be made as we enter a new golden age for classical minimal surface theory. Through our telling of the story of the classification of minimal planar domains, we hope to pass on to the general mathematical public a glimpse of the intrinsic beauty of classical minimal surface theory and our own perspective of what is happening at this historical moment in a very classical subject.

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Section: Definition 221 Any Isolated Point E ∈ E(m ) Is Called a Simentioning

confidence: 99%

The classical theory of minimal surfaces

Meeks¹,

Pérez²

2011

Bull. Amer. Math. Soc.

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“…In particular, it is area-minimizing, and then it is stable. Hence, by Main Theorem in [6], we have uniform curvature estimates for points far from the boundary of Σ n . In particular, we get uniform curvature estimates for Σ n in a neighborhood of α 3 .…”

Section: Proposition 2 Let E Be a Compact Convex Disk In Rmentioning

confidence: 99%

Periodic Minimal Surfaces in Semidirect products

Menezes

2013

J. Aust. Math. Soc.

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“…By the curvature estimates for stable H -surfaces given in [11], the norms of the second fundamental forms of the n are uniformly bounded from above at points which are at intrinsic distance at least one from their boundaries. Since the boundaries of the n leave every compact subset of * , for each compact set of * , the norms of the second fundamental forms of the n are uniformly bounded for values n sufficiently large and such a bound does not depend on the chosen compact set.…”

Section: The Proof Of Theorem 11mentioning

confidence: 99%