Before revision:  We use our improved algorithm for maximum flow in  k-apex graphs , together with additional insights, to obtain an O(k^3 n \\textrm{polylog (nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks. This improves by a factor of k^2 over the fastest algorithm previously known for this problem Wang, SODA 2019 .
After revision:  We use our improved algorithm for maximum flow in k-apex graphs , together with additional insights, to obtain an O(k^3 n \\textrm{polylog  Wang, SODA 2019 .