Course Information


Professor Rod Gover, The University of Auckland

Course Synopsis:

Lecture 1: Conformal problems in Riemannian geometry, pseudo-Riemannian geometry, and
mathematical physics.

Beginning with the notation etc in (peudo-)Riemannan geometry, this lecture will then cover some motiving problems including constructing invariants, invariant differential operators, curvature prescription, and conformal compactification.

Lecture 2: Conformal geometry, tractor calculus, and the geometry of scale.

We define conformal manifolds and construct on these the basic conformally invariant calculus. We then explain how(pseudo-) Riemannian objects should be treated in this picture.

Lecture 3: Hypersurfaces and their geometry

We develop the basic calculus for hypersurfaces in (pseudo-)Riemannian and conformal manifolds arriving at the usual curvature quantities such as the second fundamental form, the mean curvature etc. We explain the place of these in conformal geometry. We see how to use conformal geometry to describe e.g. minimal, CMC, and totally umbilic hypersurfaces.

Lecture 4: The geometry of conformal infinity and boundary calculus

The use of conformally compact manifolds (following the initial ideas of Fefferman-Graham) as a tool in conformal geometry is explained and demonstrated. Motivated by this the tools developed in the earlier lectures are applied to study the boundary at infinity of conformally compact manifolds.

Lecture 5: Higher Willmore invariants and energies

The final application is to understand how the Willmore energy and its functional gradient (with respect to variation of embedding) fit into the picture and then the generalisation of this to higher dimensions is explained as well the link between these objects and Q-curvatures.


Background Reading: S. Curry,  and A.R. Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, 74pp, in Asymptotic Analysis in General Relativity, Eds. Daude, Hafner, Nicolas,  London Mathematical Society Lecture Note Series, Cambridge U Press. ISBN: 9781316649404 or arXiv:1412.7559

Then for the later parts: 

A.R Gover, A. Waldron,  Boundary calculus for conformally compact manifolds, {\em Indiana University Mathematics Journal}, {\bf 63} (2014) , 119–163. Arxiv version: arXiv:1104.2991

A.R Gover, A. Waldron, Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem, arXiv:1506.02723 

Download the slides from the final lecture: Higher Willmore invariants and energies – via boundary calculus

Lecturer Information

Curvature in Conformal Geometry

Professor Rod Gover, The University of Auckland

Rod Gover is a Professor of Mathematics at the University of Auckland, Auckland, New Zealand. His primary research focus is on conformal, CR, and more generally Cartan differential geometries and the application of these to problems in other areas of mathematics and mathematical physics.

Professor Gover gained his DPhil from the University of Oxford in 1989 and held positions and ARC Postdoctoral and QEII Fellowships at the University of Adelaide, the University of Newcastle, and QUT before moving to Auckland.  He has held visiting positions and memberships at a number of Institutes and Universities including the IAS Princeton, MSRI Berkeley, the IHES Paris, Elie Cartan Universite Henri Poincare, Nancy, and the ANU.