What is the shape of a uniformly massive object that generates a gravitational
potential equivalent to that of two equal point-masses? If the weight of each point-mass is
sufficiently small compared to the distance between the points then the answer is a pair of balls
of equal radius, one centered at each of the two points, but otherwise it is a certain domain of
revolution about the axis passing through the two points. The existence and uniqueness of such
a domain is known, but an explicit parameterization is known only in the plane where the region
is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving
explicitly this inverse potential problem.
We study singular Hermitian metrics on vector bundles. There are two main
results in this paper. The first one is on the coherence of the higher rank analogue of multiplier
ideals for singular Hermitian metrics defined by global sections. As an application, we show
the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose
standard approximations are not Nakano semipositive. The aim of the second main result is to
determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for
example, certain rank 2 bundles on elliptic curves.