## Complete Embedded Minimal Surfaces of Finite Total Curvature (1994)

Citations: | 60 - 9 self |

### BibTeX

@MISC{Hoffman94completeembedded,

author = {David Hoffman and Hermann Karcher},

title = {Complete Embedded Minimal Surfaces of Finite Total Curvature},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

230 | A survey of minimal surfaces - Osserman - 1989 |

96 |
The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature
- Fischer-Colbrie, Schoen
- 1980
(Show Context)
Citation Context ... Carmo [1].) If the area of the Gaussian image satisfies A(N(Σ)) < 2π, then S is stable. (Here A(N(Σ)) is the area of the Gaussian image disregarding multiplicities); iii) (Fischer-Colbrie and Schoen =-=[23]-=-, do Carmo and Peng [18], Pogorelov [61].) If S is complete, S is stable if and only if S is a plane; iv) (Fischer-Colbrie � [22], Gulliver [25], Gulliver and Lawson [26].) Index (Σ) < ∞ if an only if... |

91 |
Gesammelte Mathematische Abhandlungen, vol 1
- Schwarz
- 1890
(Show Context)
Citation Context ...act D ⊂ Σ. The following theorem summarizes the results of several authors relating index to stability. Theorem 7.1 Let X: Σ →s3 be a conformal minimal immersion of an orientable surface. i) (Schwarz =-=[67]-=-.) If the Gaussian image of Σ lies in a hemisphere, S is stable; ii) (Barbosa and do Carmo [1].) If the area of the Gaussian image satisfies A(N(Σ)) < 2π, then S is stable. (Here A(N(Σ)) is the area o... |

87 |
Uniqueness, symmetry, and embeddedness of minimal surfaces
- Schoen
- 1983
(Show Context)
Citation Context ...he ends and residues of the complex differential of the height function. In Section 3, we present the few global rigidity theorems that are known. (Theorems 3.1 and 3.4, due to Lopez-Ros [51], Schoen =-=[66]-=- and Costa [16].) We present a proof (in Section 3.1) of the Lopez-Ros theorem, which states that a complete minimal surface of genus zero and finite total curvature must be the plane or the catenoid.... |

83 |
A duality theorem for Willmore surfaces
- Bryant
- 1984
(Show Context)
Citation Context ...d inequality (7.5) cannot be replaced in general by an equality. Nayatani (private communication) has pointed out to us that an example written down by kusner [44], Rosenberg-Toubiana [65], R. Bryant =-=[6, 7]-=- with 4 flat ends, genus 0, and Gauss map of degree d = 3, gives strict inequality. On any surface the support function u = X · N satisfies Lu = 0, where L is defined in (7.2), and because the ends ar... |

53 |
On complete minimal surfaces with finite Morse index in 3manifolds
- Fischer-Colbrie
- 1985
(Show Context)
Citation Context ...ork. However, the extent to which finite topology implies finite total curvature is discussed in Section 6. Section 7 discusses the index of stability of a complete minimal surface. The basic results =-=[22, 26, 25]-=-, the equivalence of finite index and finite total curvature, and the fact that the index is completely defined by the Gauss map, are discussed. There is, as yet, no known relationship between embedde... |

46 |
Construction of Minimal Surfaces, Surveys in Geometry 1989/90
- Karcher
- 1989
(Show Context)
Citation Context ...ing all of the symmetries and end-behavior. See Figure 2. This was the first known complete minimal torus with finite total curvature and one end. We will present this construction, following Karcher =-=[43]-=-, Barbosa and Colares [2] and Thayer [69, 70].son , Enneper’s surface is given by the data g = z dh = zdz (2.17) the sphere minus the point at infinity. There are no nontrivial closed cycles, so there... |

41 |
Meeks III. The topology of complete minimal surfaces of finite total Gaussian curvature
- Jorge, H
- 1983
(Show Context)
Citation Context ... pj ∈ M k. We will refer to X(Dj) = Ej as an end of M, and define, for fixed R > 0, where S 2 denotes the unit sphere insSR,j = {q ∈ S 2 |Rq ∈ Ej} (2.15) 3 . Theorem 2.3 (Gackstatter [24] Jorge-Meeks =-=[42]-=-) Let SR,j be as defined in (2.15). i) For each j = 1, . . . , r, SR,j converges smoothly as R → ∞ to a great circle, covered an integral number of times; ii) Let dj be the multiplicity of the great c... |

37 |
On embedded complete minimal surfaces of genus zero
- López, Ros
- 1991
(Show Context)
Citation Context ...th rates of the ends and residues of the complex differential of the height function. In Section 3, we present the few global rigidity theorems that are known. (Theorems 3.1 and 3.4, due to Lopez-Ros =-=[51]-=-, Schoen [66] and Costa [16].) We present a proof (in Section 3.1) of the Lopez-Ros theorem, which states that a complete minimal surface of genus zero and finite total curvature must be the plane or ... |

35 |
III, The strong halfspace theorem for minimal surfaces
- Hoffman, Meeks
- 1990
(Show Context)
Citation Context ...edded examples, flat ends of order two do not occur. It is not known if they can occur. 2.3.3 The Halfspace Theorem We present here a proof of Theorem 2.4 (The Halfspace Theorem for Minimal Surfaces, =-=[40]-=-) A complete, prop3 erly immersed, nonplanar minimal surface in is not contained in any halfspace. Our proof requires the use of the Maximum Principle for Minimal Surfaces: If S1 and S2 are two connec... |

34 |
Les surfaces à courbures opposées et leurs lignes géodesiques
- Hadamard
(Show Context)
Citation Context ...βj, whose lengths approach a minimum for all such curves, cannot diverge. Because K < 0 except at isolated points of M, a unique length-minimizing geodesic βj exists in the homotopy class 22sof � βj (=-=[27]-=-). Each geodesic bounds an end-representative that is topologically an annulus. By intersecting a planar end with a large sphere or a catenoid end with a plane orthogonal to the limit normal and suffi... |

34 |
Schrödinger operators associated to a holomorphic map
- Montiel, Ros
- 1990
(Show Context)
Citation Context ...� A are computed in the metric ds 2 N. � (| � ∇f| 2 − 2f 2 )d � A , The associated differential operator is � L = −( � ∆+2). Eigenfunctions of � ∆ can be defined by the standard variational procedure =-=[54, 71]-=-. The spectrum of � ∆ is discrete, infinite and nonnegative and we may think of index (Φ) as the number of eigenvalues of � ∆ that are strictly less than two. 80sIt is natural to try to compute the in... |

33 |
Complete minimal surfaces
- Lawson
(Show Context)
Citation Context ...sms corresponding to the symmetries of a Euclidean rectangle. Remark 4.1 The most symmetric of these Riemann surfaces are also the conformal models for one series of Lawson’s minimal surfaces in S 3 (=-=[47]-=-). They can be found by extending Plateau solutions, since the symmetry-diagonals of the quadrilaterals, the ones from the π/k-angles, are great-circle arcs of length π/2 in S 3 . The birdcage model s... |

32 |
Meeks III. Embedded minimal surfaces of finite topology
- Hoffman, H
- 1990
(Show Context)
Citation Context ...punctured torus Theorem) [16]. If S has genus one and three ends, it is one of the surfaces M1,x in Theorem 3.3. Statement 1) follows from Theorem 2.4 or Proposition 2.5 v). Statement 4) is proved in =-=[36]-=-. We will give a full proof of 2) in Section 3.1. � Corollary 3.1 The only complete embedded minimal surfaces with KdA ≥ −8π are the plane and the catenoid (with total curvature 0 and −4π respectively... |

27 |
A maximum principle at infinity for minimal surfaces and applications
- Langevin, Rosenberg
- 1988
(Show Context)
Citation Context ...at Xλ(Di)∩Xλ(Dj) = φ for λ sufficiently close to λ0. In case αi = αj and Xλ0(Di) and Xλ0(Dj) are asymptotic to ends with the same growth rate, we appeal to the Maximum Principle at Infinity proved in =-=[52]-=-, which states that the distance between these embedded annular ends is finite; i.e., they are not asymptotic at infinity. Thus, again, Xλ(Di) ∩ Xλ(Dj) = ∅ for λ sufficiently close to λ0. We may concl... |

23 |
Global geometry of extremal surfaces in three-space
- Kusner
- 1988
(Show Context)
Citation Context ...nvariant γ and R�u · ν = 0. This motivates defining the torque of a closed curve γ on S as the ∂S vector-valued quantity. Definition 2.6 � T orque0(γ) = γ X ∧ ν . Torque was introduced by Kusner � in =-=[45, 46]-=-. From (2.26) it follows that the component of torque in the �u direction is R�u · ν. If we move the origin from 0 to W ∈ 3 and let X� be γ the position vector measured from W , then � X = X − W and w... |

23 | The geometry and conformal structure of properly embedded minimal surfaces of finite topology
- Meeks, Rosenberg
- 1993
(Show Context)
Citation Context ...n a complete embedded minimal surface with finite topology? Nitsche considered ends that were fibred by embedded Jordan curves in parallel planes. These are called Nitsche ends by Meeks and Rosenberg =-=[53]-=-. For concreteness we assume that such an end A lies in the halfspace {x3 ≥ 0} with boundary a simple closed curve in x3 = 0, and that {x3 = t} ∩ A is a simple closed curve, for all t ≥ 0. Nitsche pro... |

22 |
The asymptotic behavior of properly embedded minimal surfaces of finite topology
- Hoffman, Meeks
- 1989
(Show Context)
Citation Context ... Corollary 3.1 next smallest possible total curvature for an embedded example is −12π; that it, k + r = 4. By Theorem 3.1, this can only happen when k = 1 and r = 3, and in fact it does. Theorem 3.2 (=-=[35, 36]-=-) For every k ≥ 2, there exists a complete properly embedded minimal surface of genus k − 1 with three annular ends. After suitable rotation and translation, the example of genus k − 1, which we will ... |

22 |
Some uniqueness and nonexistence theorems for embedded minimal surfaces
- Pérez, Ros
- 1993
(Show Context)
Citation Context ...(in Section 3.1) of the Lopez-Ros theorem, which states that a complete minimal surface of genus zero and finite total curvature must be the plane or the catenoid. Our proof follows that of Perez-Ros =-=[60]-=-. We also state the existence result, Theorem 3.3, for the three-ended, complete, minimal surfaces with genus k − 1 and k vertical planes of symmetry ([30]). The details of the construction of these s... |

20 | Conformal Geometry and complete minimal surfaces
- Kusner
- 1987
(Show Context)
Citation Context ...index(M) ≤ 2d − 1 (7.5) 82sThe conjectured inequality (7.5) cannot be replaced in general by an equality. Nayatani (private communication) has pointed out to us that an example written down by kusner =-=[44]-=-, Rosenberg-Toubiana [65], R. Bryant [6, 7] with 4 flat ends, genus 0, and Gauss map of degree d = 3, gives strict inequality. On any surface the support function u = X · N satisfies Lu = 0, where L i... |

20 |
Global properties of minimal surfaces
- Osserman
- 1964
(Show Context)
Citation Context ...omplete embedded minimal surface of finite total curvature has been discovered first by using the global version of the EnneperRiemann-Weierstrass representation, which is essentially due to Osserman =-=[58, 59]-=-. This involves knowledge of the compact Riemann surface structure of the minimal surface, as well as its Gauss map and other geometric-analytic data. One of our goals is to show how this is done in t... |

20 | The genus one helicoid and the minimal surfaces that led to its discovery - Hoffman, Karcher, et al. - 1993 |

19 |
Meeks III, Embedded minimal surfaces with an infinite number of ends, Invent
- Callahan, Hoffman, et al.
- 1989
(Show Context)
Citation Context ... − r) = 2πχ( � � M) = �M Σ KdA . Combining this with the previously established fact that each end has total curvature −2π gives � M � KdA = �M KdA + r(−2π) = −4π(k + r − 1) , 23swhich is (2.21). See =-=[9]-=- for more details. If S is embedded and complete with finite total curvature, then we know by Proposi3 tion 2.1 that outside of a sufficiently large compact set of , S is asymptotic to a finite number... |

17 |
Complete minimal surfaces with index one and stable constant mean curvature surfaces
- López, Ros
- 1989
(Show Context)
Citation Context ...is implies k ≤ 1. Then statements i) and ii) imply that k = 0 and d = 1. As noted above, the catenoid and Enneper’s surface are the only possibilities. Thus we have Corollary 7.1 (Corollary 9 of [54] =-=[50]-=-) A complete minimal surface with stability index equal to one must be the catenoid or Enneper’s surface. Remark 7.2 Montiel and Ros conjectured that it was possible to have a hyperelliptic Riemann su... |

16 | Adding handles to the helicoid
- Hoffman, Karcher, et al.
- 1993
(Show Context)
Citation Context ... and infinite total curvature. In 1992, we discovered, with Fusheng Wei, a complete embedded minimal surface of genus one with one end — asymptotic to the helicoid — that has infinite total curvature =-=[32, 33]-=-. The details of this contruction are outside the scope of this work. However, the extent to which finite topology implies finite total curvature is discussed in Section 6. Section 7 discusses the ind... |

16 | Construction of minimal surfaces - Karcher - 1989 |

15 |
Surfaces in conformal geometry
- Bryant
- 1988
(Show Context)
Citation Context ...d inequality (7.5) cannot be replaced in general by an equality. Nayatani (private communication) has pointed out to us that an example written down by kusner [44], Rosenberg-Toubiana [65], R. Bryant =-=[6, 7]-=- with 4 flat ends, genus 0, and Gauss map of degree d = 3, gives strict inequality. On any surface the support function u = X · N satisfies Lu = 0, where L is defined in (7.2), and because the ends ar... |

15 |
The classification of complete minimal surfaces with total curvature greater than −12π
- López
- 1992
(Show Context)
Citation Context ...r example with the symmetry of Enneper’s surface. Remark 2.3 i) The only complete minimal surfaces ins3 with total curvature −4π are the Catenoid and Enneper’s surface (Osserman [58]). Recently Lopez =-=[49]-=- proved that the unique genus-one complete minimal surface with total curvature −8π is the Chen-Gackstatter example. This result was also contained in the thesis of D. Bloss [3]. ii) The Chen-Gackstat... |

15 |
Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung. Bernhard Riemann’s gesammelte Mathematische Werke und wissenschaftlicher Nachlass
- Riemann
(Show Context)
Citation Context ... at least three. (See Remark 2.8.) However, there are complete embedded periodic minimal surfaces with flat ends where the Gauss map has degree two. The most famous of these is the example of Riemann =-=[62, 63]-=-. (See also [31], [37] and [39].) For immersed minimal surfaces, it is possible to have such an end. Consider the Weierstrass data g(z) = ((z − r)(z + r)) −1 , dh = (z 2 − r 2 )(z 2 − 1) −2 dz, r �= 1... |

14 |
A rigidity theorem for properly embedded minimal surfaces in R3
- Choi, Meeks, et al.
- 1990
(Show Context)
Citation Context ...that any intrinsic isometry of a complete embedded minimal surface of finite total curvature in 3 must extend to a symmetry of that surface; that is an isometry of 3 that leaves the surface invariant =-=[13, 36]-=-. If the surface is not embedded, this is 74snot necessarily the case. The Enneper surface and its generalizations g(z) = z k , dh =sg(z)dz, z ∈ , have intrinsic isometric rotations z → uz, |u| = 1, b... |

14 |
Index and total curvature of complete minimal surfaces
- Gulliver
(Show Context)
Citation Context ...ork. However, the extent to which finite topology implies finite total curvature is discussed in Section 6. Section 7 discusses the index of stability of a complete minimal surface. The basic results =-=[22, 26, 25]-=-, the equivalence of finite index and finite total curvature, and the fact that the index is completely defined by the Gauss map, are discussed. There is, as yet, no known relationship between embedde... |

14 |
On the stability of minimal surfaces
- Pogorelov
- 1981
(Show Context)
Citation Context ... image satisfies A(N(Σ)) < 2π, then S is stable. (Here A(N(Σ)) is the area of the Gaussian image disregarding multiplicities); iii) (Fischer-Colbrie and Schoen [23], do Carmo and Peng [18], Pogorelov =-=[61]-=-.) If S is complete, S is stable if and only if S is a plane; iv) (Fischer-Colbrie � [22], Gulliver [25], Gulliver and Lawson [26].) Index (Σ) < ∞ if an only if |K|dA < ∞. Σ Without going too much int... |

13 |
Computer-aided discovery of new embedded minimal surfaces
- Hoffman
- 1987
(Show Context)
Citation Context ... 3.1 for pictures of the Mk.) Remark 3.1 The example with k = 2 was found by Celso Costa in 1982. He proved it was complete and had embedded ends of the specified type. See Costa [14, 16] and Hoffman =-=[28, 29]-=-. The surfaces Mk each lie in a family of embedded minimal surfaces. Theorem 3.3 ([30]. See Section 4) For every k ≥ 2, there exists a one parameter family, Mk,x, x ≥ 1, of embedded minimal surfaces o... |

13 |
Meeks III,Minimal surfaces based on the catenoid
- Hoffman, H
- 1990
(Show Context)
Citation Context ...However, there are complete embedded periodic minimal surfaces with flat ends where the Gauss map has degree two. The most famous of these is the example of Riemann [62, 63]. (See also [31], [37] and =-=[39]-=-.) For immersed minimal surfaces, it is possible to have such an end. Consider the Weierstrass data g(z) = ((z − r)(z + r)) −1 , dh = (z 2 − r 2 )(z 2 − 1) −2 dz, r �= 1, on S 2 − {±1, ∞}. The surface... |

11 |
On the size of a stable minimal surface in R
- BARI3OSA, CARMO
- 1976
(Show Context)
Citation Context ...ability. Theorem 7.1 Let X: Σ →s3 be a conformal minimal immersion of an orientable surface. i) (Schwarz [67].) If the Gaussian image of Σ lies in a hemisphere, S is stable; ii) (Barbosa and do Carmo =-=[1]-=-.) If the area of the Gaussian image satisfies A(N(Σ)) < 2π, then S is stable. (Here A(N(Σ)) is the area of the Gaussian image disregarding multiplicities); iii) (Fischer-Colbrie and Schoen [23], do C... |

11 |
Computer graphics tools for the study of minimal surfaces
- Callahan, Hoffman, et al.
- 1988
(Show Context)
Citation Context ...e a well-defined finite total curvature surface. This parameter search is typically done by computer using a combination of relatively simple numerical routines and relatively complex graphics tools (=-=[8, 41]-=-). In many cases a full theoretical analysis, as is done here in Section 4 for the three-ended surfaces of Theorem 3.3 has yet to be carried out. Moreover, solution of the period problem does not at a... |

11 |
Elliptische und hyperelliptische Funktionen und vollständige Minimalflächen vom Enneperschen Typ
- Chen, Gackstatter
- 1982
(Show Context)
Citation Context ...n the boundary planes of the smallest slab are tangent to M, a contradiction, unless the slab has zero thickness. But this means M is a plane. 2.2 The example of Chen-Gackstatter Chen and Gackstatter =-=[10, 11]-=- were the first to construct an example by explicitly solving the period problem. In retrospect, and from our point of view, what they did was put a handle in Enneper’s surface, making it a genus-one ... |

11 |
Ouevres Mathématiques de Riemann
- Riemann
- 1898
(Show Context)
Citation Context ... at least three. (See Remark 2.8.) However, there are complete embedded periodic minimal surfaces with flat ends where the Gauss map has degree two. The most famous of these is the example of Riemann =-=[62, 63]-=-. (See also [31], [37] and [39].) For immersed minimal surfaces, it is possible to have such an end. Consider the Weierstrass data g(z) = ((z − r)(z + r)) −1 , dh = (z 2 − r 2 )(z 2 − 1) −2 dz, r �= 1... |

11 |
A cylindrical type complete minimal surface in a slab of R3
- Rosenberg, Toubiana
- 1987
(Show Context)
Citation Context ...tured unit disk with an essential singularity at the origin for which (2.7) (2.8) provide a well-defined embedding. Toubiana and Rosenberg have given examples which are well-defined, but notsembedded =-=[64]-=-. We also remark that on , g(z) = exp(F (z)), and dh = dz, where F (z) is entire, gives a complete, immersed, simply-connected minimal surface whose Gauss map has an essential singularity at infinity.... |

10 |
Minimal Surfaces I. Grundlehren der mathematischen Wissenschaften 295
- Dierkes, Hildebrandt, et al.
- 1992
(Show Context)
Citation Context ...mmon, near which S1 lies on 28sFigure 2.3.3 The vertically translated half-catenoid C in the proof of Theorem 2.4. one side of S2, then S1 = S2 or one is a subset of the other. We refer the reader to =-=[17]-=- for proofs. In the simple case that S2 is a plane, the Maximum Principle states that a minimal surface S = S1 that lies locally on one side of its tangent plane TpS = S2 must in fact be planar. This ... |

10 |
Higher Genus Minimal Surfaces by Growing Handles Out of a Catenoid
- Wohlgemuth
- 1991
(Show Context)
Citation Context ...t ends and two catenoid ends [73]. (See Figure 5.0.1.) 2) A one parameter family deforming the surfaces above through surfaces with four catenoid ends. Eventually these surfaces cease to be embedded. =-=[72, 4, 73, 74]-=-. 3) The surfaces in 2) with symmetric tunnels through their waist planes [72, 4]. These surfaces have k ≥ 2 vertical planes of symmetry and genus 3(k − 1). The family begins with a surface with two f... |

8 |
vision number and stability of complete minimal surfaces
- Index
- 1990
(Show Context)
Citation Context ...neralizations to higher symmetry (See Remark 4.2). Proposition 7.3 (Montiel and Ros [54]) Suppose φ: Σ → S 2 is a degree d ≥ 1 holomorphic map. If all the branch value of φ lie on a great circle Choe =-=[12]-=-, proved a weaker version of this proposition. Theorem 7.4 ([54]) The index of stability of: i) the n-noid is 2n − 3; ii) the Chen-Gackstatter surface is 3. index(Φ) = 2d − 1 . (7.4) Proof. From Secti... |

8 |
The genus one helicoid and the minimal surfaces that led to its discovery
- Ho®man, Karcher, et al.
- 1993
(Show Context)
Citation Context ... and infinite total curvature. In 1992, we discovered, with Fusheng Wei, a complete embedded minimal surface of genus one with one end — asymptotic to the helicoid — that has infinite total curvature =-=[32, 33]-=-. The details of this contruction are outside the scope of this work. However, the extent to which finite topology implies finite total curvature is discussed in Section 6. Section 7 discusses the ind... |

7 |
Meeks III, One-parameter families of embedded complete minimal surfaces with finite topology. GANG preprint in preparation
- Hoffman, H
(Show Context)
Citation Context ...atenoid. Our proof follows that of Perez-Ros [60]. We also state the existence result, Theorem 3.3, for the three-ended, complete, minimal surfaces with genus k − 1 and k vertical planes of symmetry (=-=[30]-=-). The details of the construction of these surfaces are presented in Section 4. We include here the estimation of the parameters that solve the period problem when k > 2. The values of the parameters... |

7 |
C.: A characterization of the catenoid
- Nitsche
- 1962
(Show Context)
Citation Context ... assume that such an end A lies in the halfspace {x3 ≥ 0} with boundary a simple closed curve in x3 = 0, and that {x3 = t} ∩ A is a simple closed curve, for all t ≥ 0. Nitsche proved Proposition 6.1 (=-=[57]-=-) A Nitsche end, all of whose level curves are star shaped, is a catenoidal end. In particular, it has finite total curvature. A Nitsche end necessarily has the conformal type of a punctured disk. Cer... |

7 |
Eigenvalue estimates with applications to minimal surfaces
- Tysk
- 1987
(Show Context)
Citation Context ...� A are computed in the metric ds 2 N. � (| � ∇f| 2 − 2f 2 )d � A , The associated differential operator is � L = −( � ∆+2). Eigenfunctions of � ∆ can be defined by the standard variational procedure =-=[54, 71]-=-. The spectrum of � ∆ is discrete, infinite and nonnegative and we may think of index (Φ) as the number of eigenvalues of � ∆ that are strictly less than two. 80sIt is natural to try to compute the in... |

5 |
Elliptische Funktionen und vollständige Minimalflächen
- Bloß
- 1989
(Show Context)
Citation Context ...[58]). Recently Lopez [49] proved that the unique genus-one complete minimal surface with total curvature −8π is the Chen-Gackstatter example. This result was also contained in the thesis of D. Bloss =-=[3]-=-. ii) The Chen-Gackstatter construction has been generalized in two different ways. First, one can add more handles. Chen and Gackstatter did this themselves in the genus-two case[11]. Computationally... |

5 |
Minimal surfaces in S 2m (1) with extra eigenfunctions
- Ejiri, Kotani
(Show Context)
Citation Context ...n Theorem 3.2. For 2 ≤ k ≤ 38, index(Mk) = 2k + 1 . The Gauss map of Mk has degree k + 1, so equality holds in (7.5) for these surfaces. For genus 0 surfaces Montiel and Ros [54] and Ejiri and Kotani =-=[20]-=- independently proved the following result. Theorem 7.6 Formula (7.4) holds for a generic complete, genus zero, minimal surface of total curvature −4πd. In general, index(M) ≤ 2d − 1 for such a surfac... |

5 |
Über die Dimension einer Minimalfläche und zur Ungleichung von St
- Gackstatter
- 1976
(Show Context)
Citation Context ...d neighborhood of pj ∈ M k. We will refer to X(Dj) = Ej as an end of M, and define, for fixed R > 0, where S 2 denotes the unit sphere insSR,j = {q ∈ S 2 |Rq ∈ Ej} (2.15) 3 . Theorem 2.3 (Gackstatter =-=[24]-=- Jorge-Meeks [42]) Let SR,j be as defined in (2.15). i) For each j = 1, . . . , r, SR,j converges smoothly as R → ∞ to a great circle, covered an integral number of times; ii) Let dj be the multiplici... |

5 |
Morse index of complete minimal surfaces
- Nayatani
- 1992
(Show Context)
Citation Context ...vative of the Weierstrass ℘−function, geometrically normalized, on the square torus. The geometric normalization does not change the index. He subsequently improved that result: Theorem 7.5 (Nayatani =-=[56]-=-) Let Mk be the Hoffman-Meeks surface of genus k − 1, described in Theorem 3.2. For 2 ≤ k ≤ 38, index(Mk) = 2k + 1 . The Gauss map of Mk has degree k + 1, so equality holds in (7.5) for these surfaces... |