�ber Fl�chen mit einer Weingartenschen Ungleichung

Münzner, Hans

doi:10.1007/bf01111354

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…This proves that the quasi-CMC 3 Smooth embedded spheres satisfying the Alexandrov inequality (A.2) condition (1.2) in Theorem 1.1 cannot be weakened to its natural limit, given by (2.1). These examples are similar to a construction in [39]. Then, we can glue α with its reflection across the z = z(b) line in the x, z-plane, and rotate this curve around the z-axis to obtain an embedded rotational sphere in R 3 , which can obviously be constructed with C ∞ regularity by choosing κ(s) adequately.…”

Section: A Appendix: Examplesmentioning

confidence: 82%

“…In the real analytic case, Theorem 1.1 was previously known, after a theorem of Voss and Münzner (see Satz III in [39]): Any real analytic immersed sphere in R 3 satisfying (κ 1 − c)(κ 2 − c) ≤ 0 for some c > 0 is a sphere of radius 1/c. The analyticity assumption in this result cannot be removed, by the examples given in Lemma A.2, or in Panina [42].…”

Section: And Equality Holds Only Ifmentioning

confidence: 97%

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A quasiconformal Hopf soap bubble theorem

2022

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We show that any compact surface of genus zero in $${\mathbb {R}}^3$$ R 3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon’s quasiconformal Bernstein theorem. The result generalizes, among others, Hopf’s theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.

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Section: A Appendix: Examplesmentioning

confidence: 82%

Section: And Equality Holds Only Ifmentioning

confidence: 97%

A quasiconformal Hopf soap bubble theorem

2022

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“…In the real analytic case, Theorem 1.1 was previously known, after a theorem of Voss and Münzner (see Satz III in [38]): Any real analytic immersed sphere in R 3 satisfying (κ 1 − c)(κ 2 − c) ≤ 0 for some c > 0 is a sphere of radius 1/c. The analyticity assumption in this result cannot be removed, by the examples given in Lemma A.2, or in Panina [41].…”

Section: Discussion Sharpness and Consequences Of The Resultsmentioning

confidence: 97%

“…This proves that the quasi-CMC condition (1.2) in Theorem 1.1 cannot be weakened to its natural limit, given by (2.1). These examples are similar to a construction in [38]. Then, we can glue α with its reflection across the z = z(b) line in the x, z-plane, and rotate this curve around the z-axis to obtain an embedded rotational sphere Σ in R 3 , which can obviously be constructed with C ∞ regularity by choosing κ(s) adequately.…”

Section: A Appendix: Examplesmentioning

confidence: 83%

A quasiconformal Hopf soap bubble theorem

Gálvez¹,

Mira²,

Tassi³

2021

Preprint

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We show that any compact surface of genus zero in R 3 that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon's quasiconformal Bernstein theorem. The result generalizes, among others, Hopf's theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of solutions to linear elliptic equations with discontinuous coefficients.

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“…However, if we slightly weaken the ellipticity condition g < 0 to the degenerate elliptic case g ≤ 0, the umbilics of a Weingarten surface satisfying κ 2 = g(κ 1 ) fail to be isolated in general, see [29,19]. Specifically, there exist examples of such degenerate elliptic Weingarten equations where one can bifurcate from a totally umbilic sphere meeting the rotation axis, to create nonround, degenerate elliptic, rotational Weingarten spheres.…”

Section: The Non-elliptic Casementioning

confidence: 99%

Elliptic Weingarten surfaces: singularities, rotational examples and the halfspace theorem

Fernández¹,

Mira²

2022

Preprint

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We show by phase space analysis that there are exactly 17 possible qualitative behaviors for a rotational surface in R 3 that satisfies an arbitrary elliptic Weingarten equation W (κ 1 , κ 2 ) = 0, and study the singularities of such examples. As global applications of this classification, we prove a sharp halfspace theorem for general elliptic Weingarten equations of finite order, and a classification of peaked elliptic Weingarten spheres with at most 2 singularities. In the case that W is not elliptic, we give a negative answer to a question by Yau regarding the uniqueness of rotational ellipsoids.

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�ber Fl�chen mit einer Weingartenschen Ungleichung

Cited by 8 publications

References 13 publications

A quasiconformal Hopf soap bubble theorem

A quasiconformal Hopf soap bubble theorem

A quasiconformal Hopf soap bubble theorem

Elliptic Weingarten surfaces: singularities, rotational examples and the halfspace theorem

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