Minimal Cones and the Spherical Bernstein Problem, I.

Hsiang, Wu-Yi

doi:10.2307/2006954

Cited by 62 publications

(55 citation statements)

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“…As k → ∞, Σ 2,2,k converges to the cone over the Clifford Torus. A similar phenomena was observed by Hsiang, who constructed a family of embedded O(2) × O(2)-invariant minimal surfaces in S 4 , each homeomorphic to S 3 (Theorems 1 and 2, [11]). In particular, the examples from our Theorem 1, part (1) illustrate a lack of smooth compactness for free boundary minimal surfaces in B n (1) for n > 3, in marked contrast to the compactness theorem [5] proved by Fraser and Li in dimension three.…”

Section: Introductionsupporting

confidence: 70%

“…Hsiang developed these methods, which he called "Equivariant Differential Geometry" to carry out various constructions (cf. [10], [11], [12], [13]). In particular, Hsiang proved the existence of non-equatorial embedded minimal hyperspheres in S n for several n > 3 (settling the so-called "Spherical Bernstein Conjecture"), and the existence of infinitely many noncongruent, closed, embedded minimal surfaces in S n for n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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Free boundary minimal surfaces in the unit ball with low cohomogeneity

Freidin

Gulian

McGrath

2016

Proc. Amer. Math. Soc.

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Abstract. We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers (m, n) such that m, n > 1 and m + n ≥ 8, we construct a free boundary minimal surface Σm,n ⊂ B m+n (1) invariant under O(m) × O(n). When m + n < 8, an instability of the resulting equation allows us to find an infinite family {Σ m,n,k } k∈N of such surfaces. In particular, {Σ 2,2,k } k∈N is a family of solid tori which converges to the cone over the Clifford Torus as k goes to infinity. These examples indicate that a smooth compactness theorem for Free Boundary Minimal Surfaces due to Fraser and Li does not generally extend to higher dimensions.For each n ≥ 3, we prove there is a unique nonplanar SO(n)-invariant free boundary minimal surface (a "catenoid") Σn ⊂ B n (1). These surfaces generalize the "critical catenoid" in B 3 (1) studied by Fraser and Schoen.

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Section: Introductionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Free boundary minimal surfaces in the unit ball with low cohomogeneity

Freidin

Gulian

McGrath

2016

Proc. Amer. Math. Soc.

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“…The beginning case of n = 3 was proved to be affirmative (2, 3) before the above problem was proposed. In two recent papers of the first author (4,5), 18 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C.…”

Section: Introductionmentioning

confidence: 99%

On the construction of nonequatorial minimal hyperspheres in S ⁿ (1) with stable cones in R ^{n
+1}

Hsiang¹,

Sterling²

1984

Proc. Natl. Acad. Sci. U.S.A.

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“…These results are closely related to some fundamental problems in algebraic K-theory listed by Hsiang in ref. 3.…”

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confidence: 99%

“…For each path q in B of the form q(t) = p o q(f(x, t)), [2] where x E E, t E [0, 1] and 4i: E x [0, 1] -* E is the projection onto the first factor, there exists a path q(t) in some leaf L of one of the foliations ?i such that (i) d(q(t), q(t)) < 6, for all t E [0, 1]; (ii) the diameter of q(t), measured in L, is less than a. [3] Let Pa" 6N(E; p) denote the subspace of P(E) consisting of all pseudo-isotopies that are (a, 6)-controlled relative to p.…”

mentioning

confidence: 99%