Lemma 22. Every bounded-above plurisubharmonic function on M is constant, including a function which is not assumed continuous.

Allow the constant -∞ , and otherwise let v be such a function. The compact-ball maximum m(r)=_ B_g(o,r) v is finite and nondecreasing. It is convex as a function of r . To see this, fix 0<r_1<r_2 and compare on the closed annulus with the logarithmic chord H(r)=m(r_1) +(m(r_2)-m(r_1))/( r_2- r_1) ( r- r_1). If its slope is zero the desired comparison is immediate from the definition of m(r_2) . For positive slope, a positive maximum of v-H(_g(o,·)) would occur in the annulus interior. At such a point choose a minimizing radial segment. The length index form with linear parallel test fields in the J -radial direction gives an upper Hessian bound for distance. More explicitly, let T be the final radial unit vector, let T(ℓ) denote its parallel extension, and let r be the length upper support obtained from these trial paths. At the endpoint of a segment of length r the real Hessian satisfies Hess r(T,T) +Hess r(JT,JT) ≤-1 r^3 ∫_0^r ℓ^2(T(ℓ),JT(ℓ),JT(ℓ),T(ℓ))\,dℓ<0. Indeed the radial term is -1/r^2 , while the linear test field in the JT direction gives 1/r^2 minus the displayed curvature integral. This is the strictly negative Levi derivative on the radial complex line. This support argument applies also at cut points: smoothly vary the endpoint and use trial paths along the chosen minimizing segment. Their lengths give a smooth upper support with the same index-form bound. Since H is increasing, it produces a smooth upper test for v at the hypothetical positive maximum, with a negative Levi derivative. A plurisubharmonic function admits no such upper test, whether or not it is continuous. Hence v≤H on the annulus, proving the claimed convexity. A nondecreasing convex function of r∈ which is bounded above must be constant. Upper semicontinuity implies _ r0 m(r)=v(o) , so v attains its global maximum at o . The plurisubharmonic maximum principle on connected M shows that v is constant.