On the Hausdorff dimension of the Sierpinski gasket with respect to the harmonic metric

The spectral dimension and the Hausdorff dimension of the Sierpinski gasket are known to be different. Kigami proved that the Hausdorff dimension d(h) with respect to the harmonic metric is less than or equal to the spectral dimension and conjectured equality. We give estimates of dh which are strictly less than the spectral dimension and therefore disprove this conjecture.