Cohomogeneity-Three HyperK\"ahler Metrics on Nilpotent Orbits
Differential Geometry
2007-05-23 v1
Abstract
Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperK\"ahler metric admitting a hyperK\"ahler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperK\"ahler potential is explicitly calculated when O is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperK\"ahler metrics with G-invariant K\"ahler potentials if and only if g is sp(3), su(6), so(7), so(12) or e_7 and otherwise is the unique G-invariant hyperK\"ahler metric with G-invariant K\"ahler potential.
Cite
@article{arxiv.math/0406009,
title = {Cohomogeneity-Three HyperK\"ahler Metrics on Nilpotent Orbits},
author = {Martin Villumsen},
journal= {arXiv preprint arXiv:math/0406009},
year = {2007}
}