Algebraic Minimal Surfaces Research Articles

(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

We introduce the real minimal surfaces family by using the Weierstrass data (ζ−m,ζm) for ζ∈C, m∈Z≥2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Distribution value of algebraic curves and the Gauss maps on algebraic minimal surfaces

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in [Formula: see text] with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field.

Degree 3 Algebraic Minimal Surfaces in the 3-Sphere

We give a local analytic characterization that a minimal surface in the 3-sphere S3 ⊂▪4 defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (Characterization of order 3 algebraic immersed minimal surfaces of S3, Geom. Dedicata 129 (2007), 23–34).

The Gauss map of pseudo‐algebraic minimal surfaces in R4

AbstractIn this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number of the Gauss map for pseudo‐algebraic and algebraic minimal surfaces in Euclidean four‐space and give a kind of unicity theorem (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Gauss map of pseudo-algebraic minimal surfaces

We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a new proof of Fujimoto's theorem for this class, which not only simplifies the proof but also reveals the geometric meaning behind it.

Formulae for null curves deriving from elliptic curves

Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C 3 and thus an algebraic minimal surface in R 3 . We derive simple formulae for the coordinate functions of such a null curve.

Algebraic curves and the Gauss map of algebraic minimal surfaces

In this paper, we first establish a second main theorem for algebraic curves into the n-dimensional projective space. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in R m with finite total curvature, as well as the uniqueness problem.

Non-existence of regular algebraic minimal surfaces of spheres of degree 3

In this paper we prove that if $$ M \subset S^{3} $$ is a closed minimal surface, then, $$ M \ne S^{3} \cap f^{{ - 1}} (0) $$ , for any homogeneous polynomial f of degree 3 with 0 a regular value of the function $$ f| {_{{S^{3} }} :S^{3} \to { \bf R}} . $$ .

On the totally ramified value number of the Gauss map of minimal surfaces

In this paper, we define the totally ramified value number (TRVN) of the Gauss map of a complete minimal surface, and prove that there exist algebraic minimal surfaces that have the TRVN equal to $2.5$.

Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean 3-space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

On a Variety of Algebraic Minimal Surfaces in Euclidean 4-Space

On a Variety of Algebraic Minimal Surfaces in Euclidean 4-Space