Related Experiment Videos
Conformal invariants associated to a measure
Sun-Yung A Chang1, Matthew J Gursky, Paul Yang
1Department of Mathematics, Princeton University, Princeton, NJ 08540, USA. chang@math.princeton.edu
Summary
This study defines conformally invariant Ricci and scalar curvatures for Riemannian manifolds with a smooth measure. Methods are adapted to construct conformally covariant operators and generalize the Einstein-Hilbert action.
Area of Science:
- Differential Geometry
- Mathematical Physics
Background:
- Riemannian manifolds equipped with a smooth measure present unique geometric challenges.
- Understanding conformal invariants is crucial for studying geometric properties under conformal transformations.
Purpose of the Study:
- To define and study conformal invariants of Riemannian manifolds with a smooth measure.
- To introduce conformally invariant Ricci and scalar curvatures.
- To construct conformally covariant operators and explore variational problems.
Main Methods:
- Adaptation of Fefferman-Graham and Graham et al. methods.
- Development of natural definitions for Ricci and scalar curvatures.
- Construction of families of conformally covariant operators.
Main Results:
- Natural definitions for conformally invariant Ricci and scalar curvatures are established.
- Families of conformally covariant operators are constructed.
- A generalization of the Einstein-Hilbert action is considered.
Conclusions:
- The study successfully defines conformally invariant curvatures and operators for measure-equipped Riemannian manifolds.
- The work extends existing methods to a broader class of geometric spaces.
- This research opens avenues for further investigation into variational problems in conformal geometry.