Surfaces In Sphere Research Articles

Minimal surfaces inS3foliated by circles

We study minimal surfaces in the unit sphere S 3 that are one-parameter families of circles. Minimal surfaces in R 3 foliated by circles were first investigated by Riemann, and a hundred years later Lawson constructed examples of such surfaces in S 3 . We prove that in S 3 only two types of minimal surfaces are foliated by circles crossing the principal lines at a constant angle. The first type of surfaces are foliated by great circles that are bisectrices of the principal lines, and we show that these are the examples of Lawson. The second type, which are new in the literature, are families of small circles, and the circles are principal lines. We give a constructive formula for these surfaces and an application to the theory of minimal foliated semisymmetric hypersurfaces in R 4 .

Discrete Minimal Surface Algebras

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.

Minimal surfaces, Hopf differentials and the Ricci condition

We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials, which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture for exceptional minimal surfaces in odd dimensional spheres or in S 4m .

Contact angle for immersed surfaces in [formula omitted

In this paper we introduce the notions of contact angle and of holomorphic angle for immersed surfaces in odd dimensional spheres. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in S 5 with the two angles constant. This classification gives a 2-parameter family of minimal tori of S 5 . Also, we give an alternative proof of the classification of minimal Legendrian surfaces of S 5 with constant Gaussian curvature.

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.

Hypertension in Adults Across the Age Spectrum. Current Outcomes and Control in the Community

In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method.In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce finite order rotational symmetries. We also prove, for the symmetric space which is the target space of the conformal Gauss map of Willmore surfaces in spheres, the longstanding conjecture of the existence of meromorphic invariant potentials for the conformal Gauss maps of all compact Willmore surfaces in spheres. We also illustrate our results by some concrete examples.

Spinning membranes

We present new solutions of the classical equations of motion of bosonic (matrix-)membranes. Those relating to minimal surfaces in spheres provide spinning membrane solutions in AdSp×Sq, as well as in flat space–time. Nontrivial reductions of the BMN matrix model equations are also given.

Normalized potentials of minimal surfaces in spheres

We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere S2n in terms of certain normal curvatures of the surface. When the Hopf differential is zero the potential embodies a system of first order equations governing the directrix curve of a superminimal surface in the twistor space of the sphere. We construct a birational map from the twistor space of S2n into ℂPn(n+1)/2. In general, birational geometry does not preserve the degree of an algebraic curve. However, we prove that the birational map preserves the degree, up to a factor 2, of the twistor lift of a superminimal surface in S6 as long as the surface does not pass through the north pole. Our approach, which is algebro-geometric in nature, accounts in a rather simple way for the aforementioned first order equations, and as a consequence for the particularly interesting class of superminimal almost complex curves in S6. It also yields, in a constructive way, that a generic superminimal surface in S6 is not almost complex and can achieve, by the above degree property, arbitrarily large area.

Plücker formulae for the orthogonal group

Plücker formulae for horizontal curves in SO(m)-flag manifolds are derived. These formulae are seen to generalise the usual Plücker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

Homogeneous minimal surfaces in Euclidean spheres with flat normal connections

We classify, up to congruence, homogeneous minimal surfaces in Euclidean spheres with flat normal connections. The parameter varieties in the space of contact invariants of E. Cartan are computed for all codimensions.

Ruled surfaces in the sphere

Ruled surfaces in the sphere

On a complexity of a surface in 3-sphere

On a complexity of a surface in 3-sphere

The spherical two-piece property and tight surfaces in spheres

The spherical two-piece property and tight surfaces in spheres

Minimal immersions of surfaces in Euclidean spheres

Minimal immersions of surfaces in Euclidean spheres