Conformal Geometry and Q-Curvature Analysis
Summary
Conformal geometry examines properties of shapes that remain invariant under angle-preserving transformations, focusing on the study of conformal metrics and their associated curvature invariants. In even dimensions, the Q-curvature emerges as a natural generalisation of the two-dimensional Gauss curvature, encapsulating fourth-order information about a manifold’s intrinsic geometry. Analysis of Q-curvature typically involves fourth-order elliptic operators—most notably the Paneitz and higher-order GJMS operators—and leads to challenging prescribing and compactness problems for nonlinear partial differential equations. Work in this field combines geometric analysis, spectral theory and global techniques to address questions of existence, uniqueness and stability of conformal metrics with prescribed Q-curvature. Developments in gluing constructions, variational methods and curvature flows have broadened the scope of applications, from rigorous uniformisation results on closed manifolds to insights into renormalised volume expansions in mathematical physics.
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Conformal Geometry and Q-Curvature Analysis publication trend
The graph below shows the total number of articles in conformal geometry and q-curvature analysis across all publications each year (not limited to Nature Index journals).
Technical terms
Conformal metric: A Riemannian metric obtained by scaling a reference metric by a positive smooth function, preserving angles but not lengths.
Q-curvature: A scalar curvature invariant in even dimensions associated with fourth-order conformal Laplacians, generalising Gauss curvature.
Paneitz operator: A fourth-order, conformally covariant differential operator acting on functions, central to Q-curvature equations in dimension four and higher.
GJMS operators: A family of higher-order, conformally invariant differential operators extending the Laplacian and Paneitz operator to arbitrary even order.
n-Laplacian: A nonlinear generalisation of the Laplace operator defined by div(|∇u|ⁿ⁻²∇u), appearing in the study of n-superharmonic functions and conformal geometry in higher dimensions.
References
- Equivariant Solutions to the Optimal Partition Problem for the Prescribed Q-Curvature Equation. The Journal of Geometric Analysis (2024).
- Optimal control for the Paneitz obstacle problem. ESAIM Control Optimisation and Calculus of Variations (2023).
- On n-superharmonic functions and some geometric applications. Calculus of Variations and Partial Differential Equations (2021).
- Connected sum construction of constant Q-curvature manifolds in higher dimensions. Differential Geometry and its Applications (2015).
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