[Paper Review] Semigenerated step-3 Carnot algebras and applications to sub-Riemannian perimeter

Enrico Le Donne, Terhi Moisala|arXiv (Cornell University)|Apr 18, 2020

Geometric Analysis and Curvature Flows7 references4 citations

TL;DR

This paper introduces the concept of semigenerated Carnot algebras—Carnot groups where the semigroup generated by each horizontal half-space is a vertical half-space—and provides a complete characterization for step-3 groups: they are semigenerated if and only if they have no Engel-type quotients. The key contribution is a strong rectifiability result for sets of finite intrinsic perimeter in such groups, extending prior work by Franchi, Serapioni, and Serra-Cassano.

ABSTRACT

This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call \emph{semigenerated} those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type $(\Diamond)$ and which generalizes the previous notion of type $(\star)$ defined by M. Marchi. As an application, we get that in type $ (\Diamond) $ groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano.

Motivation & Objective

• To identify the class of Carnot groups in which the only sets with constant intrinsic normal are vertical half-spaces.
• To understand the algebraic conditions under which horizontal half-spaces generate vertical half-spaces under semigroup operations.
• To provide a complete characterization of semigeneration in step-3 Carnot groups using quotient group analysis.
• To generalize the notion of type $(\star)$ groups to type $(\Diamond)$ groups for broader applicability to semigeneration criteria.
• To establish strong rectifiability results for sets of finite intrinsic perimeter in semigenerated groups.

Proposed method

• Define semigenerated Carnot algebras via the semigroup generation of horizontal half-spaces into vertical half-spaces.
• Introduce Engel-type quotients as minimal counterexamples to semigeneration, particularly in step-3 groups.
• Use algebraic structure theory to characterize semigeneration in step-3 groups by excluding Engel-type quotients.
• Define a new class of Carnot groups of type $(\Diamond)$, generalizing M. Marchi's type $(\star)$ groups.
• Apply the theory to derive rectifiability results for sets of finite perimeter via intrinsic normal behavior.
• Leverage the intrinsic geometry of Carnot groups to analyze the propagation of normal vectors under group operations.

Experimental results

Research questions

• RQ1 Which Carnot groups admit only vertical half-spaces as sets with constant intrinsic normal vector field?
• RQ2 What algebraic condition on a Carnot group ensures that the semigroup generated by a horizontal half-space is a vertical half-space?
• RQ3 For step-3 Carnot groups, when is the absence of Engel-type quotients equivalent to semigeneration?
• RQ4 How can the notion of type $(\star)$ be generalized to a broader class of Carnot groups to ensure semigeneration?
• RQ5 What rectifiability properties hold for sets of finite intrinsic perimeter in semigenerated Carnot groups?

Key findings

• A Carnot group of step 3 is semigenerated if and only if it has no Engel-type quotient algebra.
• The introduction of Engel-type groups provides a minimal obstruction to semigeneration, serving as the smallest counterexamples.
• The class of type $(\Diamond)$ Carnot groups generalizes type $(\star)$ groups and ensures semigeneration under certain conditions.
• In type $(\Diamond)$ groups and in step-3 groups without Engel-type quotients, sets of finite intrinsic perimeter are rectifiable in the sense of Franchi, Serapioni, and Serra-Cassano.
• The semigroup generation property fully characterizes the geometric rigidity of intrinsic normal fields in these groups.
• The paper establishes a strong rectifiability result for finite perimeter sets, extending known results to a broader class of Carnot groups.