A curing membrane intended for the manufacture of a tire comprising a crown region, a bead region, and a sidewall region, said membrane having, at rest, a contact surface of bulging cylindrical shape which extends about and along a central axis from a first foot forming a north pole, as far as a second foot forming a south pole, and passing via an equatorial line centered on the central axis and comprised in an equatorial plane normal to said central axis, which divides the contact surface into a northern hemisphere and a southern hemisphere, said curing membrane being provided with a drainage structure which comprises air-discharge paths cut into the contact surface to a depth equal to or greater than 0.05 mm and referred to as “channels”, said curing membrane being wherein, at least in the northern hemisphere, and preferably symmetrically in the northern hemisphere and in the southern hemisphere, there may be defined a first parallel, distinct from the equatorial line and corresponding to the intersection of the contact surface with a first plane parallel to the equatorial plane, and a second parallel which is situated between the first parallel and the corresponding foot and which corresponds to the intersection of the contact surface with a second plane parallel to the equatorial plane so that the drainage structure comprises a first drainage region corresponding to that portion of the contact surface that is delimited by the equatorial line and the first parallel, and wherein the channels have a first mean depth, a second drainage region which corresponds to that portion of the contact surface that is delimited by the first parallel and the second parallel, and wherein the channels, which communicate with the channels of the first drainage region, have a second mean depth, strictly greater than the first mean depth of the channels of the first drainage region, and a third drainage region, which corresponds to that portion of the contact surface that is delimited by the second parallel and the foot that forms the corresponding pole, and wherein the channels, which communicate with the channels of the second drainage region, have a third mean depth, strictly greater than the second mean depth of the channels of the second drainage region.