Space-like maximal surfaces and time-like minimal in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We interested the case where surface changes type from space-like to at a given non-degenerate null curve. consider this phenomenon its interesting connection 2-dimensional fluid mechanics expository article.

It is well-known that space-like maximal surfaces and time-like minimal in Lorentz--Minkowski $3$-space $\boldsymbol{R}^{3}_{1}$ have singularities general. They are both characterized as zero mean curvature surfaces. We interested the case where singular set consists of a light-like line, since this has not been analyzed before. As continuation previous work by authors, we give first example family such which change type across line. corollary, also obtain hypersurfaces...

CMC-1 trinoids (i.e. constant mean curvature one immersed surfaces of genus zero with three regular embedded ends) in hyperbolic 3-space $H^{3}$ are irreducible generically, and the ones have been classified. However, reducible case has not yet fully treated, so here we give an explicit description that includes case.

We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature 1 with singularities and elliptic ends in de Sitter 3-space. An immersed end a surface is "elliptic end" if the monodromy representation at diagonalizable eigenvalues unit circle. also give necessary sufficient condition equality to hold, process doing this we derive determining when are embedded.

The first author studied spacelike constant mean curvature one (CMC-1) surfaces in the de Sitter 3-space S 3 1 when have no singularities except within some compact subsets and are of finite total on complement this subset.However, there many CMC-1 whose singular sets not compact.In fact, such examples already appeared construction trinoids given by Lee last via hypergeometric functions.In paper, we improve Osserman-type inequality author.Moreover, shall develop a fundamental framework that...

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2\pi/n)$-rotation symmetry respect to its axis. In this paper, we show that corresponding maximal $f_n$ Lorentz-Minkowski $\boldsymbol{R}^3_1$ an analytic extension $\tilde f_n$ as properly embedded mean curvature surface. changes type into time-like (minimal)

The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for this kind obtained. As a consequence, some existence uniqueness results Möbius strips Klein bottles one end proved.

This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and in particular constant mean curvature (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit H. Wu. There already exist number of other introductions this method, but all them require higher degree mathematical sophistication from the reader than needed here. The authors' goal create exposition would be readily accessible beginning graduate student, even highly motivated undergraduate...

The first author studied spacelike constant mean curvature one (CMC-1) surfaces in de Sitter 3-space when the have no singularities except within some compact subset and are of finite total on complement this subset. However, there many CMC-1 whose singular sets not compact. In fact, such examples already appeared construction trinoids given by Lee last via hypergeometric functions. paper, we improve Osserman-type inequality author. Moreover, shall develop a fundamental framework that allows...

We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces genus with either two elliptic ends or hyperbolic in de Sitter 3-space. An end surface is an 'elliptic end' (respectively 'hyperbolic end') if monodromy matrix at diagonalizable eigenvalues unit circle reals). Although numerical, types are mathematically determined.

In this paper, we consider complete non-catenoidal minimal surfaces of finite total curvature with two ends. A family such least absolute is given. Moreover, obtain a uniqueness theorem for from its symmetries.

We investigate the relation between quadrics and their Christoffel duals on one hand, certain zero mean curvature surfaces Gauss maps other hand. To study timelike minimal of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves type change for real isothermic mixed causal turn out to be aligned with line net.

It is classically known that the only zero mean curvature entire graphs in Euclidean 3-space are planes, by Bernstein's theorem. A surface Lorentz-Minkowski $\boldsymbol{R}^3_1$ called of mixed type if it changes causal from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found two not planes. As far as authors know, these examples were without singularities. this paper, we construct several families real analytic $3$-space. The mentioned above lie one classes.

We introduce the de Sitter Schwarz map for hypergeometric differential equation as a variant of classical map. This turns out to be dual hyperbolic map, and it unifies various maps studied before. An example is also studied.

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2\pi/n)$-rotation symmetry respect to its axis. In this paper, we show that corresponding maximal $f_n$ Lorentz-Minkowski $\boldsymbol{R}^3_1$ an analytic extension $\tilde f_n$ as properly embedded mean curvature surface. changes type into time-like (minimal)