  We investigate the Laplacian Abelian gauge on the sphere S^4 in the
background of a single `t Hooft instanton. To this end we solve the eigenvalue
problem of the covariant Laplace operator in the adjoint representation. The
ground state wave function serves as an auxiliary Higgs field. We find that the
ground state is always degenerate and has nodes. Upon diagonalisation, these
zeros induce toplogical defects in the gauge potentials. The nature of the
defects crucially depends on the order of the zeros. For first-order zeros one
obtains magnetic monopoles. The generic defects, however, arise from zeros of
second order and are pointlike. Their topological invariant is the Hopf index
S^3 -> S^2. These findings are corroborated by an analysis of the Laplacian
gauge in the fundamental representation where similar defects occur. Possible
implications for the confinement scenario are discussed.
