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A Hilbert--Mumford criterion for nilsolitons

Yoshinori Hashimoto

Abstract

We give an algebraic criterion for a nilpotent real Lie algebra and prove that it provides a necessary and sufficient condition for the associated nilpotent Lie group to admit left-invariant Ricci solitons, called nilsolitons. As an application of this result, we generalise Nikolayevsky's criterion for the existence of nilsolitons to nilpotent Lie algebras without nice bases. We further prove a modified version of the Taketomi--Tamaru conjecture for nilpotent Lie groups which gives an obstruction to the existence of nilsolitons.

A Hilbert--Mumford criterion for nilsolitons

Abstract

We give an algebraic criterion for a nilpotent real Lie algebra and prove that it provides a necessary and sufficient condition for the associated nilpotent Lie group to admit left-invariant Ricci solitons, called nilsolitons. As an application of this result, we generalise Nikolayevsky's criterion for the existence of nilsolitons to nilpotent Lie algebras without nice bases. We further prove a modified version of the Taketomi--Tamaru conjecture for nilpotent Lie groups which gives an obstruction to the existence of nilsolitons.
Paper Structure (12 sections, 24 theorems, 84 equations)

This paper contains 12 sections, 24 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Related topics
  4. Preliminaries on nilsolitons
  5. Variational formalism
  6. Pre-Einstein derivation
  7. Riemannian symmetric space
  8. Kempf--Ness energy
  9. Proof of the results
  10. Hilbert--Mumford criterion
  11. Generalisation of Nikolayevsky's criterion
  12. Modified Taketomi--Tamaru conjecture

Key Result

Theorem 1.1

Let $(N, \textsl{g}_N)$ be a homogeneous nilmanifold and $(\mathfrak{n} , \mu )$ be the corresponding nilpotent Lie algebra with the inner product $\langle , \rangle$, with respect to which the pre-Einstein derivation is self-adjoint. Then, $N$ admits a left-invariant Ricci soliton if and only if $\

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 49 more