On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Danijela Damjanovic; James Tanis; Zhenqi Jenny Wang

doi:10.3934/dcds.2020164

Article Contents

2020, Volume 40, Issue 12: 6747-6766. Doi: 10.3934/dcds.2020164

This issue Previous Article Time dependent center manifold in PDEs Next Article On the regularity of the Green current for semi-extremal endomorphisms of

$ \mathbb{P}^2 $

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

1.
Royal Institute of Technology, Stockholm, Sweden
2.
The MITRE Corporation, McLean, VA 22102, USA
3.
Michigan State University, East Lansing, MI 48824, USA

^* Corresponding author: Danijela Damjanovic
^* Corresponding author: Danijela Damjanovic

† The author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions, or viewpoints expressed by the author. ©2019 The MITRE Corporation. ALL RIGHTS RESERVED

Received: June 2019

Revised: January 2020

Early access: March 2020

Published: August 2020

The first author is supported by Swedish Research Council grant VR-2015-04644. The third author is supported by NSF grant DMS-1346876. Approved for Public Release; Distribution Unlimited. Public Release Case Number 19-2033

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Abstract

We define globally hypoelliptic smooth $ \mathbb R^k $ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $ k = 1 $, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous $ \mathbb R^k $ actions ($ k\ge 2 $) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $ \mathbb R^2 $ actions on homogeneous spaces $ G/\Gamma $, with at least one quasi-unipotent generator, where $ G = SL(n, \mathbb R) $. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.

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Mathematics Subject Classification: Primary: 37C15, 37C85, 37D20.

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