On graphic Bernstein type results in higher codimension

Let be a minimal submanifold of that can be represented as the graph of a smooth map . We apply a formula that we derived in the study of mean curvature flow to obtain conditions under which must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of .

     

On the cycle map for torsion algebraic cycles of codimension two

Oblatum 7-XI-1990 &; 25-III-1991     

Categories of algebraic sets

The category of algebraic sets is defined in a straightforward way for any algebraic theory . It is a concrete, complete and cocomplete category dually equivalent to a full reflective subcategory of the category of -algebras. For the algebraic theory of commutative algebras over a field K, we get the algebraic sets over K.     

On disjunctions of algebraic sets in completely simple semigroups

A semigroup S is called an equational domain if any finite union of algebraic sets over S is algebraic. We give some necessary and su?cient conditions for a completely simple semigroup to be an equational domain.     

The isoperimetric problem in higher codimension

We consider three generalizations of the isoperimetric problem to higher codimension and provide results on equilibrium, stability, and minimization.     

Bernstein type theorems for higher codimension

We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improves previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds. Received: September 25, 1998 / Accepted: October 23, 1998     

Rigidity of isometric immersions of higher codimension

Given an isometric immersionf:M ⁿ → ℝ ^N into Euclidean space, we provide sufficient conditions onf so that any 1-regular isometric immersion ofM ⁿ into ℝ ^N+1 is necessarily obtained as a composition off with a local isometric immersion ℝ ^N ⊃U → ℝ ^N+1 . This result has several applications.     

Affine algebraic sets relative to an algebraic theory

For an arbitrary algebraic theory T and a prescribed T-algebraL, the geometrical category AfAlgSet(L) of affine algebraic sets over the affine lineL is build up. It is proved to be dually equivalent to the category FcAlg(L) of functional T-algebras overL. The Galois theory ofL is defined and a Galois duality established. These geometrical categories are characterized up to an equivalence of categories.     

Minimal rambling sets with codimension one dominated splittings

In this paper,we study the dynamics of minimal rambling sets with codimension one dominatedsplittings     

Anisotropic curve shortening flow in higher codimension

We consider the evolution of parametric curves by anisotropic mean curvature flow in ?ⁿ for an arbitrary n?2. After the introduction of a spatial discretization, we prove convergence estimates for the proposed finite‐element model. Numerical tests and simulations based on a fully discrete semi‐implicit stable algorithm are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Anisotropic mean curvature flow in higher codimension

We present recent progress in the study of the anisotropic mean curvature flow in higher codimension. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The exterior Plateau problem in higher codimension

We prove existence theorems for two-dimensional, noncompact, complete minimal surfaces in ℝⁿ of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within a bounded distance from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity flat ends as well as of surfaces with polynomial-type nonflat ends. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Homotopy properties of algebraic sets

In this paper there are announced several homotopy theorems on algebraic subsets of complex projective spaces. Some of the theorems generalize and refine results of the preceding note of the author (RZhMat, 1978, 5A531). It is also asserted that any -dimensional complex affine algebraic set has the homotopy type of a finite -dimensional cellular space. A generalization is given, in two versions, of a theorem of Bart-Larsen (RZhMat, 1973, 7A587).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 67–72, 1979.     

Harmonic measure on sets of codimension larger than one

We introduce a new notion of a harmonic measure for a d-dimensional set in ${\mathbb{R}}^n$ with $d<n?1$, that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the d-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.