Hiroyasu Satoh

My research area is differential geometry, particularly in harmonic manifolds and asymptotically harmonic manifolds.
A harmonic manifold is a Riemannian manifold where the volume density function, expressed in polar coordinates centered at any point $p$, depends only on the distance $r=d(p,\cdot)$ from the center.
This condition is equivalent to the value of the mean curvature of the geodesic sphere $S_p(r)$ being a function of the radius $r$, independent of the center $p$. On the other hand, an asymptotically harmonic manifold is a Riemannian manifold without conjugate points, where the mean curvature of the holospheres,
which is obtained as the limit of geodesic spheres, has a common constant value.

Regarding harmonic manifolds, the Lichnerowicz conjecture suggested that "harmonic manifolds are only two-point homogeneous spaces", but counterexamples in the non-compact case (Damek-Ricci spaces) have been found.
However, (1) it remains unresolved whether there exist harmonic manifolds other than two-point homogeneous spaces or Damek-Ricci spaces, and (2) examples of asymptotically harmonic manifolds that are not harmonic manifolds have not been found.

We defined harmonic manifolds of hypergeometric type as spaces where the eigenfunctions of the Laplace operator are represented by hypergeometric functions. This constitutes a new class of harmonic manifolds, including two-point homogeneous spaces with negative curvature (rank one symmetric spaces of non-compact type) and Damek-Ricci spaces.
This definition, equivalent to the mean curvature of the geodesic sphere being expressible in a certain form, leads to several geometric properties.

Future research is expected to yield new discoveries regarding non-compact harmonic manifolds and asymptotically harmonic manifolds.
Additionally, new insights into the characterization of two-point homogeneous spaces and the construction of new examples of harmonic manifolds are anticipated.