A Course in Minimal Surfaces

Colding, Tobias Holck; Minicozzi, William P.

doi:10.1090/gsm/121

Cited by 281 publications

(298 citation statements)

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“…As a preliminary remark, we observe that the improved decay estimate |A 0 (x)| ≤ o(1)|x| −1 (which follows from the proof of Lemma 4.4 together with (4.1)) implies via a standard graphicality lemma (as in Chapter 2 of [CM11]) that for any σ large enough Σ 0 can be described, in the Euclidean annulus of radii σ and 2σ as a graph over a coordinate plane. Specifically, for any such σ and there exists a plane Π = Π (σ) in coordinates {x} and a suitably smooth function v = v (σ) : Π → R whose graph coincides with Σ 0 in the aforementioned ambient annulus.…”

Section: Isometric Embedding In the Minkowski Spacetimementioning

confidence: 71%

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Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

Carlotto

2016

Ann. Henri Poincaré

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Abstract. In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of possibly non-compact marginally outer trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set (M, g, k) obeying the dominant energy condition is conformally diffeomorphic to either the plane C or to the cylinder A and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly this rules out the existence of trapped regions with cylindrical boundary. In the second part of the article, we restrict our attention to asymptotically flat data (M, g, k) and show that, in that setting, the existence of an unbounded, stable marginally outer trapped surface essentially never occurs unless in a very specific case, since it would force an isometric embedding of (M, g, k) into the Minkowski spacetime as a space-like slice.

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Section: Isometric Embedding In the Minkowski Spacetimementioning

confidence: 71%

“…Thanks to the locally uniform bounds for these graphical components (again: as in Chapter 2 of [CM11]), one easily gets that…”

Section: Isometric Embedding In the Minkowski Spacetimementioning

confidence: 99%

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

Carlotto

2016

Ann. Henri Poincaré

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“…This argument is known as a normal variation argument, and it is standard within the literature of isoperimetric inequalities or of the calculus of variations. For just a few of many possible references which are relevant here, see [8,6,5]. In fact, we are dealing with the noise stability of multiple sets, and within Lemma 3.1 we perturb two sets A i , A j simultaneously, while leaving the remaining sets intact.…”

Section: 1mentioning

confidence: 99%

Standard simplices and pluralities are not the most noise stable

2016

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Abstract. The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ = 0 of the noise and for every prescribed measures for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell's result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.

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“…Thus, if * is homeomorphic to S 2 \{N , S}, either by combining results of Schoen [28] [23] and [7] for a survey on the minimal surface theory in R 3 . Thus, since K * is not constant, * must be either a catenoid or a helicoid.…”

Section: Classification Of Stationary Isothermic Surfaces In Rmentioning

confidence: 99%

“…See also [23] and [7] for the minimal surface theory in R 3 . Now, item (iii) above guarantees that there exists δ > 0 such that, for every x ∈ S, the connected component of B δ (x) ∩ S containing x is written as a graph of a function over the tangent plane to S at x (see [7,Lemma 2.4,p.…”

Section: Remark 41mentioning

confidence: 99%

Stationary isothermic surfaces in Euclidean 3-space

2015

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Let be a domain in R 3 with ∂ = ∂(R 3 \ ), where ∂ is unbounded and connected, and let u be the solution of the Cauchy problem for the heat equation ∂ t u = u over R 3 , where the initial data is the characteristic function of the set c = R 3 \ . We show that, if there exists a stationary isothermic surface of u with ∩ ∂ = ∅, then both ∂ and must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that ∩ ∂ = ∅ and ∂ is unbounded. To prove this result, we establish a similar theorem for uniformly dense domains in R 3 , a notion that was introduced by methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.

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A Course in Minimal Surfaces

Cited by 281 publications

References 0 publications

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

Standard simplices and pluralities are not the most noise stable

Stationary isothermic surfaces in Euclidean 3-space

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