Singularities of geometric flows and applications (SINGFA)
Hellenic Foundation for Research and Innovation (H.F.R.I)
Project Number: 2958
Geometric flows have proven to be a powerful tool in the investigation of problems in Geometry and Topology, with their most prominent success being the resolution of Thurston’s Geometrization Conjecture by Perelman in 2002-03 using Hamilton’s Ricci flow.
The overarching principle behind their application is that they deform a given geometric structure to optimal geometric structures that are easier to classify, revealing information about the underlying space. A key component of the procedure is the development of singularities and the ability of the flow to continue past them, in this search of optimal geometric structures.
This project addresses important current challenges in geometric flows and their application to Geometry and Topology. A large part of the project is concerned about Ricci flow, aiming towards a deeper understanding of the development of finite time singularities and of the existence and uniqueness of Ricci flows starting from singular initial data.
A separate major part of the project aims to apply geometric flow techniques in the realm of G2 geometry on seven dimensional manifolds. Despite the success of elliptic PDE methods in the construction of special G2 holonomy Riemannian metrics, many questions remain unanswered. Geometric flow techniques is a promising tool, however such techniques have only recently begun to be explored in the context of G2 geometry.
Team members
Theodora Bourni, University of Tennessee, Knoxville
Shubham Dwivedi, Humboldt Universität zu Berlin
Panagiotis Gianniotis (PI), National and Kapodistrian University of Athens
Spiro Karigiannis, University of Waterloo
Konstantinos Leskas (postdoc), National and Kapodistrian University of Athens
Felix Schulze, University of Warwick
Georgios Zacharopoulos (PhD student), National and Kapodistrian University of Athens
Publications
Splitting maps in Type I Ricci flows, Calc. Var. 64, 212 (2025). https://doi.org/10.1007/s00526-025-03044-z [journal, arXiv]
A G_2-Hilbert functional in G_2 geometry, arXiv:2505.06872, joint with George Zacharopoulos
Diameter bounds in 3d Type I Ricci flows, arXiv:2510.14019 dadfsdfarXiv:2510.14019adv:2510.14019adrXiv:2510.14019arXiv:2510.14019
L^1 bounds on the curvature tensor for Type I Ricci flows, joint with Konstantinos Leskas (in preparation)
Isometric flow in the presence of torsion free G_2-structures, joint with Shubham Dwivedi (in preparation)