Morse theory with the norm-square of a hyperkähler moment map

We prove that the norm-square of a moment map associated to a linear action of a compact group on an affine variety satisfies a certain gradient inequality. This allows us to bound the gradient flow, even if we do not assume that the moment map is proper. We describe how this inequality can be extended to hyperkähler moment maps in some cases, and use Morse theory with the norm-squares of hyperkähler moment maps to compute the Betti numbers and cohomology rings of toric hyperkähler orbifolds. 1