Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Closed $G_2$-Structures with Negative Ricci Curvature

Alec Payne

TL;DR

This work establishes that closed $G_2$-structures on compact manifolds cannot realize negative Ricci curvature, and extends a pinching argument to noncompact manifolds to prove a noncompact version of the $G_2$-Goldberg conjecture: a complete Einstein closed $G_2$-structure must be torsion-free and hence Ricci-flat. The key tool is a precise integral identity relating Ricci curvature, torsion, and the closed condition, which together with volume-growth analysis and Bishop–Gromov comparison yields global obstructions to complete, closed $G_2$-structures with negative or narrowly negative Ricci curvature. The results reveal strong restrictions on curvature for closed $G_2$-geometries, with additional geodesic-length bounds for incomplete cases and implications for the Laplacian flow and broader holonomy contexts.

Abstract

We study existence problems for closed $G_2$-structures with negative Ricci curvature, and we prove the $G_2$-Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed $G_2$-structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed $G_2$-structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed $G_2$-structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics.

Closed $G_2$-Structures with Negative Ricci Curvature

TL;DR

This work establishes that closed -structures on compact manifolds cannot realize negative Ricci curvature, and extends a pinching argument to noncompact manifolds to prove a noncompact version of the -Goldberg conjecture: a complete Einstein closed -structure must be torsion-free and hence Ricci-flat. The key tool is a precise integral identity relating Ricci curvature, torsion, and the closed condition, which together with volume-growth analysis and Bishop–Gromov comparison yields global obstructions to complete, closed -structures with negative or narrowly negative Ricci curvature. The results reveal strong restrictions on curvature for closed -geometries, with additional geodesic-length bounds for incomplete cases and implications for the Laplacian flow and broader holonomy contexts.

Abstract

We study existence problems for closed -structures with negative Ricci curvature, and we prove the -Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed -structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed -structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed -structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics.
Paper Structure (4 sections, 10 theorems, 48 equations)

This paper contains 4 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries on G2-structures
  4. Proofs of Main Results

Key Result

Theorem 1.1

There does not exist a closed $7$-manifold with a closed $G_2$-structure $\varphi$ such that $\mathop{\mathrm{Ric}}\nolimits(g_{\varphi})<0$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1: BryantRemarks
  • proof
  • Lemma 2.2: KarigiannisFlows09
  • Lemma 2.3: CleytonIvanovCurvatureDecomp08
  • proof
  • Lemma 3.1
  • ...and 10 more