A characterization of a standard torus in $E^3$

Shiohama, Katsuhiro; Takagi, Ryoichi

doi:10.4310/jdg/1214429643

Cited by 36 publications

(29 citation statements)

References 3 publications

(4 reference statements)

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“…In this case J^a H2 dA=2n2 is the infimum for all anchor rings, a fact first shown by Willmore. He then raises the question hinted at by both Willmore [4] and Shiohama-Takagi [3] as to whether these special anchor rings are the only unknotted tori in E3 which satisfy equation (1). Our result can be extended to give a negative answer to this question.…”

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confidence: 77%

A Global Invariant of Conformal Mappings in Space

White¹

1973

Proceedings of the American Mathematical Society

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Abstract.This paper shows that the total integral of the square of the mean curvature for a compact orientable surface in E3 is an invariant of a conformai space mapping. This result is then used to answer a problem raised by T. Willmore and B.-Y. Chen concerning embeddings of compact orientable surfaces, and in particular tori, for which this integral is a minimum.A conformai mapping on Euclidean three-space has the property that it carries spheres into spheres. Such a mapping can be decomposed into a product of similarity transformations and inversions. Let x:M2^-E3 be a C3 regular immersion of a compact orientable surface into Euclidean three-space, and let T=(XI2 H2 dA, where H is the mean curvature of the immersed surface and dA is the area element. This note proves the result that T is invariant under a conformai space mapping and applies this result to the problem mentioned by T. J. Willmore That ris invariant under similarity transformations (Euclidean motions and homotheties) is obvious. What is not so apparent is that Tis invariant under inversions. (We stipulate here that the center of inversion does not lie on the immersed surface.) It was observed locally by Blaschke [1] that the quantity (H2-K)dA is an inversion invariant, where K is the Gauss curvature.To prove this fact let the center of inversion be taken as the origin. Then, if c is the radius of inversion, the position vector x of a point on the inverse surface, corresponding to the point x on the original surface, has the direction of x and the magnitude c2/|x| where |x| denotes the length of x. Therefore, we may write x=c2x:/|x|2. If N denotes the usual surfaceReceived by the editors June 15, 1972. AMS (MOS) subject classifications (1970). Primary 53A05; Secondary 53A30, 53C45.

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confidence: 77%

A Global Invariant of Conformal Mappings in Space

White¹

1973

Proceedings of the American Mathematical Society

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“…We will denote by ψ : M → H 3 ⊂ L 4 an isometric immersion of M into H 3 , and by N its (globally defined) Gauss map. We will also denote by λ 1 , λ 2 the principal curvatures of M associated to N .…”

Section: Let Lmentioning

confidence: 99%

Complete surfaces in the hyperbolic space with a constant principal curvature

Aledo

Gálvez

2005

Mathematische Nachrichten

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In this paper we study complete orientable surfaces with a constant principal curvature R in the 3-dimensional hyperbolic space H 3 . We prove that if R 2 > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H 3 . When R 2 ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6].

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“…Using this fact, Shiohama and Takagi [18] gave a partial answer to the Willmore conjecture (cf. [12]).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, since Gaussian curvature is the product of the principal curvatures, K = λ 1 λ 2 , a flat surface can be regarded as a surface where one of the principal curvatures is identically zero. In the case of a non-zero constant, Shiohama-Takagi [18] proved the following classification theorem.…”

Section: Introductionmentioning

confidence: 99%

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Weakly Complete Wave Fronts With One Principal Curvature Constant

Honda

2016

Kyushu J. Math.

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Abstract. Murata and Umehara gave a classification of complete flat fronts in the Euclidean 3-space and proved their orientability. Here, a flat front is a flat surface (i.e., a surface where one of the principal curvatures is identically zero) with admissible singularities. In this paper, we investigate wave fronts where one of the principal curvatures is a non-zero constant. Although they are orientable in the regular surface case, there exist non-orientable examples. We classify weakly complete ones and derive their orientability.

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A characterization of a standard torus in $E^3$

Cited by 36 publications

References 3 publications

A Global Invariant of Conformal Mappings in Space

A Global Invariant of Conformal Mappings in Space

Complete surfaces in the hyperbolic space with a constant principal curvature

Weakly Complete Wave Fronts With One Principal Curvature Constant

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