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The Airborne Internet is a vision of a large-scale multihop wireless mesh network consisting of commercial passenger aircraft connected via long-range highly directional air-to-air radio links. We propose a geographic load sharing strategy to fully exploit the total air-to-ground capacity available at any given time. When forwarding packets for a given destination, a node considers not one but a set of next-hop candidates and spreads traffic among them based on queue dynamics. In addition, load balancing is performed among Internet Gateways by using a congestion-aware handover strategy. Our simulations using realistic North Atlantic air traffic demonstrate the ability of such a load sharing mechanism to approach the maximum theoretical throughput in the network.