Foundations for computer-assisted existence proofs of Calabi-Yau and G2-manifolds

• Applicant: Fabian Lehmann, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 573641031

My research is concerned with special holonomy manifolds, in particular Calabi-Yau manifolds and manifolds with holonomy G2. These are of great significance in differential geometry because they are Ricci-flat and thus examples of Einstein manifolds. Yau's proof of the Calabi conjecture establishes the existence of many Ricci-flat Kähler metrics on compact complex manifolds. However, the proof is not constructive and provides little information on the metric. For manifolds with holonomy G2 there is no general existence result, and all available constructions yield metrics which are close to a degenerate limit and manifolds with high topological complexity. In recent years, computer-assisted proofs have emerged as a promising tool to construct solutions of complicated partial differential equations. The first step of a computer-assisted proof is an analytical reduction of solving the equation to finding an ``approximate'' solution. Building on my expertise in the perturbation of special geometric structures, the goal of my research programme is to derive such a reduction for future computer-assisted existence proofs of Ricci-flat Kähler metrics and metrics with holonomy G2. In the Kähler setting, I plan to prove explicit estimates for the Laplace-Beltrami operator. This will provide a rigorous criterion for an ``approximate'' solution. Building on this, I plan to use the optimal algebraic metrics introduced by Headrick--Nassar to complete a computer-assisted proof of the Ricci-flat Kähler metric on the Fermat quartic, which is independent of Yau's proof. In the G2-setting, I plan to develop a general approach to perturb closed G2-structures with small torsion to torsion-free G2-structures. Previous work by Joyce and Platt only applies to gluing constructions. In another direction, I plan to use the heat flow of Weiss--Witt to investigate whether such a perturbation theory can be extended to G2-structures that are not closed.

• DFG Programme: Research Grants
• International Connection: Canada, United Kingdom, USA
• Cooperation Partners: Professor Michael R. Douglas; Professor Javier Gomez Serrao; Daniel Platt, Ph.D.; Professor Freid Tong