By analyzing the first variation of area, we characterize C2 area-stationary surfaces as those with mean curvature zero or constant if a volume-preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski-type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we re-fine the Bernstein type theorem given in [CHMY] and [GP] to describe all C2 entire area-stationary graphs over the xy-plane in H1.

A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the description of the singular set in [CHMY], the charac-terization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify area-stationary surfaces in H1, with or without a volume constraint, and non-empty singular set.

In particular, we deduce the following counterpart to Alexandrov unique-ness theorem in Euclidean space: any compact, connected, C2 surface in H1, area-stationary under a volume constraint, must be congruent to a rotation-ally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points.

As a consequence, we solve the isoperimetric problem in H1 assuming C2 smoothness of the solutions. Documents: Advanced Search Include Citations. Authors: Advanced Search Include Citations. Abstract Abstract. Keyphrases area-stationary surface heisenberg group h1 volume constraint first variation minkowski-type formula carnot-carathe odory distance main result volume-preserving area-stationary surface volume con-straint alexandrov unique-ness theorem mean curvature singular curve calibration argument constant mean curvature surface rotation-ally symmetric isoperimetric problem volume-preserving condition characteristic curve mean curvature zero c2 entire area-stationary graph c2 area-stationary surface c2 smoothness singular set c2 surface non-empty singular set bernstein type theorem euclidean space following counterpart.

Powered by:.These results are then used to prove a Heisenberg version of the Gaussâ€”Bonnet theorem. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. In the publication [1] there is an unfortunate computational error, which however does not affect the correctness of the main results.

### Minimal translation surfaces in the Heisenberg group Nil3

Agrachev, A. Discrete Contin. Arcozzi, N. Balogh, Z. Reine Angew. Nonlinear Anal. Bao, D. Capogna, L. Potenza Progress in Mathematics, th edn. Google Scholar. Cheng, J. Pisa Cl. Chiu, H. Partial Differ. Chousionis, V. In preparation. Citti, G.

Potential Anal. Danielli, D. Forum Math. Diniz, M. Control Syst. Prentice-Hall, Inc. Translated from the Portuguese Mathematics: Theory and Applications. Translated from the second Portuguese edition by Francis Flaherty Hladky, R. Lee, J. Graduate Texts in Mathematics, th edn.

Springer, NewYork Ni, Y. Pura Appl. Pauls, S. Download references.We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a straight ruled surface, i. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Capogna, L. In: Bass, H. Progress in Mathematics, vol.

Cheng, J-H. Pisa Cl. Google Scholar. Do Carmo M. Prentice-Hall, USA Danielli D. Indiana Univ. Danielli, D. Garofalo, N. Goldman, W. Clarendon Press, Oxford Inoguchi J-I. McDuff, D. Pauls S. Download references. You can also search for this author in PubMed Google Scholar.

Correspondence to Ioannis D.Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Huang and S. HuangS. We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively.

The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume.

The application makes a connection between CR geometry and integral geometry. View PDF on arXiv.

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Citation Type. Has PDF. Publication Type. More Filters. Research Feed. The differential geometry of curves in the Heisenberg groups. The kinematic formula in the 3D-Heisenberg group.

The existence of horizontal envelopes in the 3D-Heisenberg group. View 5 excerpts, cites background. The existence of horizontal envelopes in the 3 D-Heisenberg group Yen-Chang. The fundamental theorem of curves and classifications in the Heisenberg groups. View 4 excerpts, cites methods and background. Tubular neighborhoods in the sub-Riemannian Heisenberg groups. View 1 excerpt, cites background.Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers resulting in the "continuous Heisenberg group" or the ring of integers resulting in the "discrete Heisenberg group".

The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stoneâ€”von Neumann theorem. More generally, one can consider Heisenberg groups associated to n -dimensional systems, and most generally, to any symplectic vector space. As one can see from the term ab'the group is non-abelian.

The neutral element of the Heisenberg group is the identity matrixand inverses are given by. If a, b, care real numbers in the ring R then one has the continuous Heisenberg group H 3 R. It is a nilpotent real Lie group of dimension 3.

In addition to the representation as real 3x3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces. By Stoneâ€”von Neumann theoremthere is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character. This representation has several important realizations, or models. In the theta representationit acts on the space of holomorphic functions on the upper half-plane ; it is so named for its connection with the theta functions.

If a, b, care integers in the ring Z then one has the discrete Heisenberg group H 3 Z. It is a non-abelian nilpotent group. It has two generators.

By Bass's theoremit has a polynomial growth rate of order 4. It is a group of order p 3 with generators x,y and relations:. Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special groupsor more properly, extra special groups of exponent p.

The Heisenberg group modulo 2 is of order 8 and is isomorphic to the dihedral group D 4 the symmetries of a square. Observe that if.

The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics:.

The Heisenberg group is a simply-connected Lie group whose Lie algebra consists of matrices.

By letting e 1In particular, z is a central element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent. The exponential map evaluates to. The exponential map of any nilpotent Lie algebra is a diffeomorphism between the Lie algebra and the unique associated connectedsimply-connected Lie group. This discussion aside from statements referring to dimension and Lie group further applies if we replace R by any commutative ring A.

The corresponding group is denoted H n A. Under the additional assumption that the prime 2 is invertible in the ring Athe exponential map is also defined, since it reduces to a finite sum and has the form above i.

The unitary representation theory of the Heisenberg group is fairly simple â€” later generalized by Mackey theory â€” and was the motivation for its introduction in quantum physics, as discussed below. The motivation for this representation is the action of the exponentiated position and momentum operators in quantum mechanics.

The phase factor is needed to obtain a group of operators, since translations in position space and translations in momentum space do not commute. The Hamiltonian generators of translations in phase space are the position and momentum functions. The general abstraction of a Heisenberg group is constructed from any symplectic vector space. The Heisenberg group is a central extension of the additive group V.

Thus there is an exact sequence.To browse Academia. Skip to main content. Log In Sign Up. Download Free PDF. Minimal translation surfaces in the Heisenberg group Nil3 Geometriae Dedicata, Marian Ioan Munteanu. Jun-ichi Inoguchi. Download PDF. A short summary of this paper.

Minimal translation surfaces in the Heisenberg group Nil3. These surfaces are now referred as Scherk's minimal surfaces.

The study of translation surfaces in the Euclidean space was extended when the second fundamental form was considered as a metric on a non-developable surface. A classification is given for translation surfaces for which the second Gaussian curvature and the mean curvature are proportional [14]. When the ambient is the Minkowski 3-space, translation surfaces of Weingarten type are classified [5].

In [8], translation surfaces with vanishing second Gaussian curvature in Euclidean and Minkowski 3-space are studied. In the last decade, there has been an intensive effort to develop the theory of surfaces in homogeneous Riemannian 3-spaces of non-constant curvature. Since the discovery of holomorphic quadratic differential called generalized Hopf differential or Abresch-Rosenberg differential for constant mean curvature surfaces in 3-dimensional homogeneous Riemannian spaces with 4-dimensional isometry group, global geometry of constant mean curvature surfaces in such spaces has been extensively studied.

We refer the survey [1], [7] or lecture notes [4] and references therein. In particular, integral representation formulae for minimal surfaces in the Heisenberg group Nil 3 were obtained independently in [3,6,10].

Some fundamental examples of minimal surfaces are constructed in [11]. Translation surfaces can be defined in any 3-dimensional Lie groups equipped with left invariant Riemannian metric.To toggle the display of rejected clips on or off in the Browser window, change the top menu from Hide Rejected to Show All Clips. Your Rejected clips are either hidden or displayed, depending upon which of these two options you choose.

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