https://neckpinch.com/authors/spiro-karigiannis/New arXiv papers by Spiro Karigiannis, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2511.21055/arxiv:2511.21055Wed, 26 Nov 2025 00:00:00 GMTSpiro Karigiannis, Sébastien Picard, Caleb Suan — We study flows of G₂-structures guided by the principle of dimensional reduction: natural geometric flows in G₂-geometry reduce to natural flows in complex geometry. Our main examples are the G₂-Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The G₂-lift of the anomaly flow deforms conformally coclosed G₂-structures. We compare the G₂-anomaly flow to the G₂-Laplacian coflow, and investigate short-time existence and fixed points.Ricci flowKähler–Ricci flowGeneralized & G₂ flows3-manifoldsKähler geometryHarmonic maps & DeTurck trick https://neckpinch.com/papers/2508.11604/arxiv:2508.11604Fri, 15 Aug 2025 00:00:00 GMTSpiro Karigiannis — The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially G₂-structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of G₂-structures. Finally, we consider geometric flows of G₂-structures. We review the basics of G₂-geometry and survey several different geometric flows of G₂-structures. In particular, we clarify in what sense STE results for the G₂ Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of G₂-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.Ricci flowGeneralized & G₂ flowsStability & uniquenessHarmonic maps & DeTurck trick