Soliton solutions for the Laplacian co-flow of some \(G_2\)-structures with symmetry
type: Item
instance of: scholarly article
title: Soliton solutions for the Laplacian co-flow of some \(G_2\)-structures with symmetry
zbMATH Open document ID: 1256.53043
author: Spiro Karigiannis
author: Benjamin McKay
author: Mao-Pei Tsui
published in: Differential Geometry and its Applications
publication date: 2012-07-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.2192
review text: Let \(M^7\) be a \(7\)-dimensional, orientable manifold and for a non-degenerate 3-forms \(\varphi\), which defines a \(G_2\)-structures on \(M^7\), denote by \(\psi = \ast_{\varphi}\varphi\) the corresponding dual \(4\)-form. The authors consider the Laplacian co-flows of \(G_2\)-structures, namely, the 1-parameter families \(\varphi_t\) of non-degenerate \(3\)-forms satisfying the equation \(\frac{\partial \psi_t}{\partial t}= - \Delta_t \psi_t\), where \(\Delta_t\) denotes the Hodge Laplacian of the \(G_2\)-structure given by \(\varphi_t\). In analogy with the Ricci flow, \(G_2\)-structures that are initial data of self-similar solutions are called solitons and are characterised by equations of the form \( - \Delta \psi = \mathcal L_X \psi + \lambda \psi\) for some vector field \(X\) and some constant \(\lambda\). The authors study the Laplacian co-flows for co-closed \(G_2\)-structures (i.e., with \(d \psi = 0\)) on warped products of an interval or a circle with a compact \(6\)-dimensional manifold \(N^6\), endowed with a nearly Kähler or Calabi-Yau metric, and look for solitons on such warped products. They determine all such solitons in the Calabi-Yau case, while, in the nearly Käbler case, they determine several special solutions and prove that the general problem reduces to a single third-order ordinary differential equation.
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C26
Mathematics Subject Classification ID: 32Q25
Mathematics Subject Classification ID: 53C10
zbMATH DE Number: 6054370
zbMATH Keywords: \(G_2\)-structures
zbMATH Keywords: Laplacian flows
zbMATH Keywords: solitons
zbMATH Keywords: geometric evolution equations
reviewed by: Andrea Spiro
MaRDI profile type: Publication
OpenAlex ID: W2165544879
Wikidata QID: Q115356713
arXiv ID: 1108.2192
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DOI: 10.1016/J.DIFGEO.2012.05.003
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description: scientific article; zbMATH DE number 6054370