The kinematic evolution of axisymmetric magnetic and electric fields is investigated numerically in Kerr geometry for a simplified Keplerian disk near a rotating black hole. In the cases investigated it is found that a magnetic field cannot be sustained against ohmic diffusion. In flat space this result is known as Cowling's antidynamo theorem. No support is found for the possibility that the gravitomagnetic dynamo effect of Khanna & Camenzind could lead to self-excited axisymmetric solutions. In practice, therefore, Cowling's antidynamo theorem may still hold in Kerr geometry, although here the original proof can no longer be applied.
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