Soliton solutions for the Laplacian co-flow of some \(G_2\)-structures with symmetry
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.
scientific article; zbMATH DE number 6054370
scientific article