The classical Fourier inversion formula writes any nice function on Euclidean space as a superposition of “plane waves”, i.e. eigenfunctions of the Laplacian which are constant on a family of parallel hyperplanes. For the n-dimensional real hyperbolic space, and more generally for symmetric spaces of noncompact type, there is an analogous Fourier transform, called the Helgason Fourier transform, which similarly allows one to write functions as superpositions of “plane waves”, where in this case the plane waves are eigenfunctions of the Laplacian which are constant on a family of “horospheres” (these are given by the family of hypersurfaces normal to the set of geodesics meeting at a common boundary point on the boundary at infinity of the symmetric space).

We generalize the Fourier transform to the purely geometric setting of harmonic manifolds. These are Riemannian manifolds where harmonic functions satisfy the mean-value property with respect to the Riemannian surface measure on geodesic spheres. Examples of such manifolds include Euclidean space, the n-dimensional hyperbolic space, and more generally rank one symmetric spaces of noncompact type. We define a Fourier transform for negatively curved harmonic manifolds and prove a Fourier inversion formula in this setting.

In later joint work with Knieper and Peyerimhoff, this Fourier inversion formula was extended to the case of harmonic manifolds of purely exponential volume growth, a class which includes all known examples of simply connected, noncompact, nonflat harmonic manifolds.

The video of this talk is available on the IISc Math Department channel.