Plurisubharmonicity of envelopes of disc functionals
on manifolds
Finnur Lárusson1 and Ragnar Sigurdsson2
1Dept. of Mathematics,Univ. of Western Ontario, London,
Ontario N6A 5B7, Canada
e-mail: larusson@uwo.ca
2Science Institute, University of Iceland, IS-107 Reykjavik,
Iceland
e-mail: ragnar@raunvis.hi.is
We show that a disc functional on a complex manifold has a
plurisubharmonic envelope if all its pullbacks by holomorphic
submersions from domains of holomorphy in affine space do and it is
locally bounded above and upper semicontinuous in a certain weak sense.
For naturally defined classes of disc functionals on manifolds, this
result reduces a property somewhat stronger than having a
plurisubharmonic envelope to the affine case. The proof uses a recent
Stein neighbourhood construction of Rosay, who proved the
plurisubharmonicity of the Poisson envelope on all manifolds. As a
consequence, the Riesz envelope and the Lelong envelope are
plurisubharmonic on all manifolds; for the former, we make use of new
work of Edigarian. The basic theory of the three main classes of disc
functionals is thereby extended to all manifolds.