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We present a formula that relates the variations of the area of extremethroat initial data with the variation of an appropriate defined massfunctional. From this expression we deduce that the first variation, with fixedangular momentum, of the area is zero and the second variation is positivedefinite evaluated at the extreme Kerr throat initial data. This indicates thatthe area of the extreme Kerr throat initial data is a minimum among this classof data. And hence the area of generic throat initial data is bounded frombelow by the angular momentum. Also, this result strongly suggests that theinequality between area and angular momentum holds for generic asymptoticallyflat axially symmetric black holes. As an application, we prove this inequalityin the non trivial family of spinning Bowen-York initial data.

Extreme throat initial data set and horizon area--angular momentum inequality for axisymmetric black holes