Scientific network to study geometric problems related to higher-order and fractional-order equations

• Applicant: Dr. Jesse Ratzkin
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 561401741

In this project we continue our ongoing investigations of the higher-order and fractional-order versions of the very successful Yamabe problem in geometric analysis, which asks for a constant scalar curvature representative in a given conformal class of Riemannian metrics. We ask the same question for other scalar-valued curvature quantities that transform according to either higher-order or fractional-order equations under a conformal change of metric. Collectively, we refer to this as the Q-curvature of a metric, of varying order. In previous work we've already constructed many new and interesting examples of both smooth and singular constant Q-curvature metrics using gluing techniques. We've investigated compactness properties and the local structure of the moduli space of constant Q-curvature metrics with isolated singularities. We've investigated the stability of the total Q-curvature functional near minimizing metrics in the smooth setting. New aspects of this project include: (i) new gluing constructions of both smooth and singular examples (ii) proving the convergence of a natural geometric flow which stationary points are the constant Q-curvature metrics (iii) classifying the rate of convergence of this flow (iv) extending our examples showing degenerate stability of the total Q-curvature functional on certain products from the integer order to the fractional order (v) progress towards classifying all solutions on a twice-punctured sphere as the so-called Delaunay solutions (vi) investigating compactness of solutions in low dimensions and blow-up sequences in high dimensions (vii) producing new and interesting foliations of an asymptotically flat end, providing interesting applications in the mathematical theory of general relativity (viii) constructing balancing diagrams associated to constant Q-curvature metrics with point singularities (ix) proving nondegeneracy of more examples with more complicated topology and (x) producing new constant Q-curvature metrics through bifurcations from the Berger spheres.

• DFG Programme: Scientific Networks