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This paper is the second part of a trilogy dedicated to the followingproblem: given spherically symmetric characteristic initial data for theEinstein-Maxwell-scalar field system with a cosmological constant $\Lambda$,with the data on the outgoing initial null hypersurface given by a subextremalReissner-Nordstrom black hole event horizon, study the future extendibility ofthe corresponding maximal globally hyperbolic development as a "suitablyregular" Lorentzian manifold.  In the first paper of this sequence, we established well posedness of thecharacteristic problem with general initial data.  In this second paper, we generalize the results of Dafermos on the stabilityof the radius function at the Cauchy horizon by including a cosmologicalconstant. This requires a considerable deviation from the strategy followed byDafermos, focusing on the level sets of the radius function instead of thered-shift and blue-shift regions. We also present new results on the globalstructure of the solution when the free data is not identically zero in aneighborhood of the origin.  In the third and final paper, we will consider the issue of mass inflationand extendibility of solutions beyond the Cauchy horizon.

communications in mathematical physics/communications in mathematical physics

On the global uniqueness for the Einstein-Maxwell-scalar field system  with a cosmological constant. Part 2: Structure of the solutions and  stability of the Cauchy horizon