Existence and nonexistence of skew branes

Existence and non-existence of skew branes

A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a non-zero Euler characteristic χ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ/4. We give a version of this result for immersed surf...

Existence and Nonexistence of Solutionsfor

Nonexistence of Skew Loops on Ellipsoids

We prove that every C closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve. A skew loop is a C closed curve without parallel tangent lines imme...

Harmonic Homeomorphisms: Existence and Nonexistence

These notes were prepared for International Workshop on Harmonic and Quasiconformal Mappings in Chennai, August 2010. They are based on several papers written jointly with Tadeusz Iwaniec, Jani Onninen, and Ngin-Tee Koh. The source papers are available from www.arxiv.org

Existence and Nonexistence of Descriptive Patterns

In the present paper, we study the existence of descriptive patterns, i. e. patterns that cover all words in a given set through morphisms and that are optimal in terms of revealing commonalities of these words. Our main result shows that if patterns may be mapped onto words by arbitrary morphisms, then there exist infinite sets of words that do not have a descriptive pattern. This answers a qu...