The convergence space of minimal USCO mappings

A convergence structure generalizing the order convergence structure on the set of Hausdorff continuous interval functions is defined on the set of minimal usco maps. The properties of the obtained convergence space are investigated and essential links with the pointwise convergence and the order convergence are revealed. The convergence structure can be extended to a uniform convergence structure so that the convergence space is complete. The important issue of the denseness of the subset of all continuous functions is also addressed.      

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

The stability of a minimal surface in a pseudo-Euclidean space

We prove that any piece of a space-like minimal surface of the form z=z(x, y), in the pseudo-Euclidean space E^3,1 with line element dx²+dy²–dz² has the greatest area among close space-like surfaces with the same boundary. For time-like surfaces the following assertion is proved. Let V² be a time-like minimal surface in E^3,1. Then for any domain on V², there exists a smooth variation in the class of time-like surfaces with fixed boundary that increases the area, and there exists a smooth variation in the same class of surfaces that decreases the area.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 41–45, 1990.     

The length of minimal filling for a five-point metric space

The length of the minimal filling of an arbitrary five-point metric space is represented as a function of lengths between points from this metric space.     

Orthogonal almost-complex structures of minimal energy

In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers exist, and in particular, we prove that the standard almost-complex structure on the round S ⁶ gives the absolute minimum for the energy. We also discuss the uniqueness of this minimum and the extension of these results to other orthogonal G-structures.     

Rigidity of minimal submanifolds in hyperbolic space

We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L ² harmonic 1-forms on M.     

The space of minimal prime ideals in a 0-distributive semilattice

LetS be a 0-distributive semilattice and be its minimal spectrum. It is shown that is Hausdorff. The compactness of has been characterized in several ways. A representation theorem (like Stone's theorem for Boolean algebras) for disjunctive, 0-distributive semilattices is obtained.     

Complete minimal hypersurfaces in complex hyperbolic space

We construct new examples of embedded, complete, minimal hypersurfaces in complex hyperbolic space, including deformations of bisectors and some minimal foliations. Received: 20 March 2000 / Revised version: 21 July 2000     

New minimal surfaces in the hyperbolic space

We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H~3 found by Mori(1981), we obtain 2-and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R~2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H~3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.     

Complete minimal hypersurfaces in quaternionic hyperbolic space

We construct new examples of embedded, complete minimal hypersurfaces in quaternionic hyperbolic space and also some minimal foliations. We introduce fans and construct analytic deformations of bisectors.     

Two-dimensional minimal surfaces in a pseudo-Euclidean space

Translated from Ukrainskii Geometricheskii Sbornik, No. 31, pp. 36–47, 1988.     

A moduli space of minimal affine Lagrangian submanifolds

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Fréchet manifold (if it is not Ø).