[Paper Review] Minimal metrics on nilmanifolds

Jorge Lauret|ArXiv.org|Nov 11, 2004

Geometric Analysis and Curvature Flows24 references23 citations

TL;DR

This paper establishes that minimal metrics on nilmanifolds—defined as left-invariant Riemannian metrics minimizing the Ricci tensor norm for fixed scalar curvature—are uniquely determined up to isometry and scaling. The key contribution is proving that such metrics exist precisely on nilpotent Lie groups that are nilradicals of standard Einstein solvmanifolds, with explicit constructions and classification of known examples across symplectic, complex, and hypercomplex structures.

ABSTRACT

A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If $N$ is endowed with an invariant symplectic, complex or hypercomplex structure, then minimal compatible metrics are also unique up to isometry and scaling. The aim of this paper is to give more evidence of the existence of minimal metrics, by presenting several explicit examples. This also provides many continuous families of symplectic, complex and hypercomplex nilpotent Lie groups. A list of all known examples of Einstein solvmanifolds is also given.

Motivation & Objective

• To establish the existence and uniqueness of minimal metrics on nilpotent Lie groups, defined as those minimizing the Ricci tensor norm under fixed scalar curvature.
• To characterize minimal metrics via equivalence to Ricci solitones, Einstein solvmanifold extensions, and derivations of the Lie algebra.
• To extend the concept of minimality to nilmanifolds equipped with invariant geometric structures—symplectic, complex, or hypercomplex—proving uniqueness of compatible minimal metrics.
• To provide a comprehensive list of known examples of nilpotent Lie groups admitting minimal metrics, including those arising from symmetric spaces, Clifford modules, and deformations.
• To clarify the relationship between minimal metrics and the geometry of Einstein solvmanifolds, particularly in the rank-one case.

Proposed method

• Define a minimal metric on a nilpotent Lie group as one minimizing $||\operatorname{ric}_{\langle\cdot,\cdot\rangle}||$ among all left-invariant metrics with the same scalar curvature.
• Use the equivalence of minimality to Ricci soliton metrics under normalized Ricci flow, ensuring isometric evolution.
• Employ the characterization that a metric is minimal if and only if $\operatorname{Ric}_{\langle\cdot,\cdot\rangle} = cI + D$ for $c \in \mathbb{R}$ and $D \in \operatorname{Der}(\mathfrak{n})$, linking to Einstein solvmanifold theory.
• Construct metric solvable extensions $\mathfrak{s} = \mathfrak{a} \oplus \mathfrak{n}$ with $\mathfrak{a}$ abelian and $\mathfrak{n}$ the nilradical, such that $\mathfrak{s}$ is Einstein if and only if $\mathfrak{n}$ admits a minimal metric.
• Apply the moment map and variational principles on the variety of nilpotent Lie algebras $\mathcal{N}$ to identify critical points of curvature functionals.
• Use the functional $F([\mu]) = \operatorname{tr}(\operatorname{Ric}_\mu^2)/||\mu||^4$ to detect Einstein-like behavior and classify nilradicals of Einstein solvmanifolds.

Experimental results

Research questions

• RQ1 Which nilpotent Lie groups admit a minimal metric, i.e., a left-invariant metric minimizing the Ricci tensor norm for fixed scalar curvature?
• RQ2 How are minimal metrics on nilmanifolds related to Einstein solvmanifolds, and under what conditions does a nilpotent Lie algebra arise as the nilradical of a standard Einstein solvmanifold?
• RQ3 Do minimal metrics compatible with invariant symplectic, complex, or hypercomplex structures on nilmanifolds exist, and are they unique up to isometry and scaling?
• RQ4 Can the existence of a minimal metric be characterized algebraically via derivations and gradings, particularly $\mathbb{N}$-gradings of the Lie algebra?
• RQ5 What is the complete list of known examples of nilpotent Lie groups admitting minimal metrics, and what structural features do they share?

Key findings

• Minimal metrics on nilpotent Lie groups are unique up to isometry and scaling, as established via variational and Ricci flow arguments.
• A nilpotent Lie group admits a minimal metric if and only if it is the nilradical of a standard Einstein solvmanifold, linking minimal metrics to Einstein geometry.
• For nilmanifolds with invariant symplectic, complex, or hypercomplex structures, minimal compatible metrics also exist and are unique up to isometry and scaling.
• The paper provides a complete list of known examples, including Iwasawa $N$-groups, $H$-type Lie groups, nilradicals of parabolic subalgebras, and families from Clifford modules and deformations.
• A $6$-step nilpotent $7$-dimensional Lie algebra forms the lowest-dimensional continuous family of nilpotent Lie algebras admitting minimal metrics.
• A $10$-dimensional $2$-step nilpotent Lie algebra with $5$-dimensional center forms a continuous curve of minimal metrics, demonstrating the richness of such structures.