Convex integration with avoidance and hyperbolic (4,6) distributions

TL;DR

This paper develops a new 'avoidance trick' within convex integration to classify complex tangent distributions, including hyperbolic (4,6) distributions, expanding the scope of the h-principle beyond classical limitations.

Contribution

It introduces a general avoidance framework and the concept of 'ample up to avoidance' for differential relations, enabling convex integration in previously intractable cases.

Findings

Proves the full h-principle for step-2 bracket-generating distributions.

Establishes a complete h-principle for hyperbolic (4,6) distributions using avoidance.

Answers a question of Eliashberg and Mishachev by providing an example of a relation ample in some directions but not all.

Abstract

This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration.