The conformal systole is related to the stable systole in a nontrivial way. The conformal systole is defined in terms of the L^2 norm in homology, while the stable systole is defined in terms of the stable norm in homology. More precisely, each of the systoles is the first successive minimum of the integer lattice in 2-dimensional homology with respect to the corresponding norm.
Unlike the relationship between the two systoles (conformal and stable), the relationship between the two norms is clear. The conformal norm is the supremum of the stable norm, over all metrics in the given conformal class. This relationship does not, however, translate into any immediate relationship between the conformal systole and the stable systole. Thus, there is still no non-trivial existence results for the stable systole, i.e. metrics with interesting asymptotic behavior of the stable systole.
For instance, the conformal systole of the sum of n copies of the complex projective plane is precisely 1, since the L^2 norm coincides with the intersection form in this signature. Therefore one also has a uniform upper bound for the stable systole of these manifolds. It is unknown whether the stable 2-systole has to tend to zero when n tends to infinity.
For the connected sum of n complex projective planes P and 1 copy of P
with the opposite orientation, we already mentioned the existence
results in the form of polynomial lower bounds for the conformal
systole. It would be interesting to apply such lower bounds to
construct metrics with nontrivial behavior of the stable 2-systole in
these manifolds. So far there are no lower bounds for the stable
systole that would distinguish the family with definite intersection
form from the family with indefinite intersection form.
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