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## Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group (2004)

Citations: | 60 - 10 self |

### Citations

1153 |
A comprehensive introduction to differential geometry. Vol
- Spivak
- 1979
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Citation Context ...herefore the total index sum of this line field is nonnegative. This index sum is equal to the Euler characteristic number of the surface Σ according to the Hopf index theorem for a line field (e.g., =-=[Sp]-=-). On the other hand, the Euler characteristic number of Σ equals 2 − 2g(Σ) where g(Σ) denotes the genus of Σ. It follows that g(Σ) ≤ 1. Q.E.D. Appendix. Basic facts in pseudohermitian geometry Let M ... |

885 | Spin geometry
- Lawson, Michelsohn
- 1989
(Show Context)
Citation Context ...and area-minimizing property In this section we will derive the second variation formula for the p-area functional (2.5) and examine the p-mean curvature H from the viewpoint of calibration geometry (=-=[HL]-=-). As a result we can prove the area-minimizing property for a pminimal graph in H1. 31We follow the notation in Section 2. We assume the surface Σ is p-minimal. Let f, g be functions with compact su... |

479 |
Topology from the Differentiable Viewpoint
- Milnor
- 1965
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Citation Context ...+d 2 = 0. Therefore in matrix form, d(N ⊥ [ ] [ ] uyx + 1 uyy 1 0 D)p0 = = . −uxx −uxy + 1 0 1 Note that det(d(N ⊥ D)p0) = 1 > 0. So the index of N ⊥ D at p0 is +1 (see, e.g., Lemma 5 in Section 6 in =-=[Mil]-=-). 17Q.E.D. Lemma 3.9. Let u ∈ C2 (Ω). Suppose |H| = o( 1 r ) (little ”o”) near an isolated singular point p0 ∈ Ω where r(p) = |p−p0|. Then there exists a small neighborhood V ⊂ Ω of p0 such that the... |

384 | The Yamabe problem
- Lee, Parker
- 1987
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Citation Context ...yley transform F: S 3 \{(0, −1)} → H1 by ζ x = Re( 1 ζ ), y = Im( 1 + ζ2 1 1 + ζ 2 ), z = 1 2 − ζ2 Re[i(1 )] 1 + ζ2 Q.E.D. where (ζ 1 , ζ 2 ) ∈ S 3 ⊂ C 2 satisfies |ζ 1 | 2 + |ζ 2 | 2 = 1 (see, e.g., =-=[JL]-=-). A direct computation shows that (7.8) ˆΘ = F ∗ (λ 2 Θ0) where λ 2 = 4[4z 2 +(x 2 +y 2 +1) 2 ] −1 (recall that Θ0 = dz+xdy−ydx is the standard contact form for H1). 37Lemma 7.3. Let Σ be a C 2 smoo... |

318 |
A Survey on Minimal Surfaces,
- Osserman
- 1969
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Citation Context ...(2.20), we have ∆t b (x, y, z) ≡ (∆tb x, ∆tb y, ∆tb z) = (0, 0, 0) (i.e., ∆t b annihilates the coordinate functions) on Σ. This is a property analogous to that for (Euclidean) minimal surfaces in R3 (=-=[Os]-=-). In general, we have ∆t b (x, y, z) = He2. We will often call the xy−plane projection of characteristic curves for a graph (x, y, u(x, y)) in H1 still characteristic curves if no confusion occurs. N... |

247 |
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un
- Pansu
- 1989
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Citation Context ...ea 2-form as follows: (2.11) Θ ∧ e 1 = Ddx ∧ dy = [(ux − y) 2 + (uy + x) 2 ] 1/2 dx ∧ dy. So the p-area of Σ (see (2.5)) identifies with the 3-dimensional spherical Hausdorff measure of Σ (see, e.g., =-=[Pan]-=-, [Pau]). At a singular point, the contact form Θ is proportional to dψ (see (2.9a), (2.9b)). Therefore ux − y = 0, uy + x = 0 describe the xy−plane projection S(u) of the singular set SΣ: (S) S(u) = ... |

223 |
Nonlinear Ordinary Differential Equations
- John, Smith
- 1986
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Citation Context ...differential equation. It is degenerate hyperbolic (on the nonsingular domain) having only one characteristic direction (note that a 2dimensional hyperbolic equation has two characteristic directions =-=[Jo]-=-). We call the integral curves of this characteristic direction the characteristic curves. We show that the p-mean curvature is the line curvature of a characteristic curve. Therefore the characterist... |

144 |
Isoperimetric and Sobolev Inequalities for Carnot-Carathéodory Spaces and the Existence of Minimal Surfaces
- Garofalo, Nhieu
- 1996
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Citation Context ... elliptic equations to approximate it. By this standard elliptic method, Pauls ([Pau]) obtained a W 1,p Dirichlet solution and made a link to the Xminimal surfaces in the sense of Garofalo and Nhieu (=-=[GN]-=-). In general the solution to the Dirichlet problem may not be unique. However, we can still establish a uniqueness theorem by making use of a structural equality of ”elliptic” type (Lemma 5.1). More ... |

135 |
Pseudo-hermitian structures on a real hypersurface
- Webster
- 1978
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Citation Context ...to tensors) given by (A.2) ∇ p.h. Z1 = ω1 1 ⊗Z1, ∇ p.h. ¯1 Z¯1 = ω¯1 ⊗Z¯1, ∇ p.h. T = 0 in which the 1-form ω1 1 is uniquely determined by the following equation with a normalization condition ([Ta], =-=[We]-=-): (A.3) (A.4) dθ 1 = θ 1 ∧ω1 1 + A 1 ¯1Θ∧θ ¯1 , ω1 1 ¯1 + ω¯1 = 0. The coefficient A 1 ¯1 in (A.3) is called the (pseudohermitian) torsion. Since h 1¯1 = 1, A¯1¯1 = h 1¯1A 1 ¯1 = A 1 ¯1. And A11 is j... |

102 |
A differential geometric study on strongly pseudo-convex manifolds,
- Tanaka
- 1975
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Citation Context ...ended to tensors) given by (A.2) ∇ p.h. Z1 = ω1 1 ⊗Z1, ∇ p.h. ¯1 Z¯1 = ω¯1 ⊗Z¯1, ∇ p.h. T = 0 in which the 1-form ω1 1 is uniquely determined by the following equation with a normalization condition (=-=[Ta]-=-, [We]): (A.3) (A.4) dθ 1 = θ 1 ∧ω1 1 + A 1 ¯1Θ∧θ ¯1 , ω1 1 ¯1 + ω¯1 = 0. The coefficient A 1 ¯1 in (A.3) is called the (pseudohermitian) torsion. Since h 1¯1 = 1, A¯1¯1 = h 1¯1A 1 ¯1 = A 1 ¯1. And A1... |

95 |
Complete minimal surface in S3,”
- Lawson
- 1970
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Citation Context ... or p-minimal surfaces of genus ≥ 2 in the standard pseudohermitian 3-sphere. Note that in the standard Euclidean 3-sphere, there exist many closed C∞ smoothly embedded minimal surfaces of genus ≥ 2 (=-=[La]-=-). On a surface in a pseudohermitian 3-manifold, we define an operator, called the tangential sublaplacian. The p-mean curvature is related to this operator acting on coordinate functions (see (2.19a)... |

81 |
The Fefferman metric and pseudo-Hermitian invariants
- Lee
- 1986
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Citation Context ...uppose ˜ Θ = λ 2 Θ, λ > 0. Then ˜ H = λ −2 (λH − 3e2(λ)). Proof. Let ˜e1 denote the characteristic field with respect to ˜ Θ. Then it follows from the definition that ˜e1 = λ −1 e1. Applying (5.7) in =-=[Lee]-=- to e1 (in our case, n = 1, Z1 = 1 2 (e1 − ie2)), we obtain (7.7) λ˜ω 1 1(˜e1) = ω1 1 (e1) − 3iλ −1 e2(λ). Note that H = ω(e1) = −iω1 1 (e1) (see the remark after (2.8)). Rewriting (7.7) in terms of H... |

40 |
Shnider: Spherical hypersurfaces in complex manifolds
- Burns, S
- 1976
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Citation Context ...there are no constant p-mean curvature or p-minimal surfaces Σ of genus ≥ 2 in M. There are many examples of spherical CR manifolds, and there have been many research studies in this direction (e.g., =-=[BS]-=-, [KT], [FG], [CT]). We speculate that Theorem E might imply a topological constraint on a spherical CR 3-manifold. The idea of the proof for Theorem E goes as follows. A spherical pseudohermitian man... |

40 |
Complete minimal surfaces
- Lawson
- 1970
(Show Context)
Citation Context ...or p-minimal surfaces of genus ≥ 2 in the standard pseudohermitian 3-sphere. Note that in the standard Euclidean 3-sphere, there exist many closed C ∞ smoothly embedded minimal surfaces of genus ≥ 2 (=-=[La]-=-). On a surface in a pseudohermitian 3-manifold, we define an operator, called the tangential sublaplacian. The p-mean curvature is related to this operator acting on coordinate functions (see (2.19a)... |

35 |
Estimates for the ∂̄b-complex and analysis on the Heisenberg group
- Folland, Stein
- 1974
(Show Context)
Citation Context ... is independent of the choice of unitary (h 1¯1 = 1) frame Z1. The real version of (A.8) reads (A.8r) ∇bf = (e1f)e1 + (e2f)e2. Next we will introduce the 3-dimensional Heisenberg group H1 (see, e.g., =-=[FS]-=-). For two points (x, y, z), (x ′ , y ′ , z ′ ) ∈ R 3 , we define the multiplication as follows: (x, y, z) ◦ (x ′ , y ′ , z ′ ) = (x + x ′ , y + y ′ , z + z ′ + yx ′ − xy ′ ). R 3 endowed with this mu... |

34 | Minimal surfaces in the Heisenberg group.
- Pauls
- 2004
(Show Context)
Citation Context ...e the Bernstein problem. Namely, we study entire p-minimal graphs (a graph or a solution is called entire if it is defined on the whole xy-plane). The following are two families of such examples (cf. =-=[Pau]-=-): (1.1) (1.2) u = ax + by + c (a plane with a,b,c being real constants); u = −abx 2 + (a 2 − b 2 )xy + aby 2 + g(−bx + ay) (a, b being real constants such that a 2 + b 2 = 1 and g ∈ C 2 ). The main r... |

17 |
Applications de l’analyse a la geometrie
- Monge
(Show Context)
Citation Context ...er development, we derive it and discuss the stability of a p-minimal surface in Sections 6 and 7. Acknowledgments. We would like to thank Ai-Nung Wang to explain what Monge wrote in his French book (=-=[Mo]-=-) about the third order equation for ruled surfaces (see Section 4). The first author would also like to thank Yng-Ing Lee to show him some basic facts in calibration geometry. We started this researc... |

16 |
On capillary free surfaces in the absence of gravity.
- Concus, Finn
- 1974
(Show Context)
Citation Context ...in Ω +\ ¯ S and hence in Ω +\S by continuity. Q.E.D. Remark. Theorem 5.2 is an analogue of Concus and Finn’s general comparison principles for the prescribed mean curvature equation (cf. Theorem 6 in =-=[CF]-=-). In [Hw2] Hwang invoked the ”tan −1 ” technique to simplify the proof of [CF]. Here we follow the idea of Hwang in [Hw2] to prove Theorem 5.2. Lemma 5.3. Let u, v ∈ C 2 (Ω) ∩ C 0 ( ¯ Ω) where Ω is a... |

10 |
On a new approach to Bernstein’s theorem and related questions for equations of minimal surface type
- Miklyukov
- 1979
(Show Context)
Citation Context ...re ∇u Tu = , we have the following structural inequality: √ 1+|∇u| 2 √ √ 1 + |∇u| 2 + 1 + |∇v| 2 (∇u − ∇v) · (Tu − Tv) ≥ |Tu − Tv| 2 2 ≥ |Tu − Tv| 2 . The above inequality was discovered by Miklyukov =-=[Mik]-=-, Hwang [Hw1], and CollinKrust [CK] independently. Here we adopt Hwang’s method to prove Lemma 5.1. Next let u ∈ C 0 ( ¯ Ω\S1), v ∈ C 0 ( ¯ Ω\S2), i.e., u, v are not defined (may blow up) on sets S1, ... |

8 |
Spherical CR-manifolds of dimension 3
- Falbel, Gusevskii
- 1994
(Show Context)
Citation Context ... constant p-mean curvature or p-minimal surfaces Σ of genus ≥ 2 in M. There are many examples of spherical CR manifolds, and there have been many research studies in this direction (e.g., [BS], [KT], =-=[FG]-=-, [CT]). We speculate that Theorem E might imply a topological constraint on a spherical CR 3-manifold. The idea of the proof for Theorem E goes as follows. A spherical pseudohermitian manifold is loc... |

7 |
Comparison principles and Liouville theorems for prescribed mean curvature equations in unbounded domains, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e serie 15 (3): 341–355
- Hwang
- 1988
(Show Context)
Citation Context ...ere Tu = ∇u√ 1+|∇u|2 , we have the following structural inequality: (∇u−∇v) · (Tu− Tv) ≥ √ 1 + |∇u|2 +√1 + |∇v|2 2 |Tu− Tv|2 ≥ |Tu− Tv|2. The above inequality was discovered by Miklyukov [Mik], Hwang =-=[Hw1]-=-, and CollinKrust [CK] independently. Here we adopt Hwang’s method to prove Lemma 5.1. Next let u ∈ C0(Ω̄\S1), v ∈ C0(Ω̄\S2), i.e., u, v are not defined (may blow up) on sets S1, S2 ⊂ Ω, respectively.... |

5 | Deformation of spherical CR structures and the universal Picard variety
- Cheng, Tsai
(Show Context)
Citation Context ...ant p-mean curvature or p-minimal surfaces Σ of genus ≥ 2 in M. There are many examples of spherical CR manifolds, and there have been many research studies in this direction (e.g., [BS], [KT], [FG], =-=[CT]-=-). We speculate that Theorem E might imply a topological constraint on a spherical CR 3-manifold. The idea of the proof for Theorem E goes as follows. A spherical pseudohermitian manifold is locally t... |

2 |
Le Problème de Dirichlet pour le equation des surfaces minimales fur des domaines non bornès
- Collin, Krust
- 1991
(Show Context)
Citation Context ...tructural inequality: √ 1+|∇u| 2 √ √ 1 + |∇u| 2 + 1 + |∇v| 2 (∇u − ∇v) · (Tu − Tv) ≥ |Tu − Tv| 2 2 ≥ |Tu − Tv| 2 . The above inequality was discovered by Miklyukov [Mik], Hwang [Hw1], and CollinKrust =-=[CK]-=- independently. Here we adopt Hwang’s method to prove Lemma 5.1. Next let u ∈ C 0 ( ¯ Ω\S1), v ∈ C 0 ( ¯ Ω\S2), i.e., u, v are not defined (may blow up) on sets S1, S2 ⊂ Ω, respectively. Let S ≡ S1 ∪S... |

2 |
inequalities method for uniqueness theorems for the minimal surface equation
- Structural
- 1997
(Show Context)
Citation Context ...S and hence in Ω +\S by continuity. Q.E.D. Remark. Theorem 5.2 is an analogue of Concus and Finn’s general comparison principles for the prescribed mean curvature equation (cf. Theorem 6 in [CF]). In =-=[Hw2]-=- Hwang invoked the ”tan −1 ” technique to simplify the proof of [CF]. Here we follow the idea of Hwang in [Hw2] to prove Theorem 5.2. Lemma 5.3. Let u, v ∈ C 2 (Ω) ∩ C 0 ( ¯ Ω) where Ω is a bounded do... |