$C^\infty$ approximations of convex, subharmonic, and plurisubharmonic functions

Convex functions on complete noncompact manifolds : differentiable structure

“…This second property is the nonsmooth analogue ofnoncriticality. In the presence of these properties, but without further appeal to convexity, the smooth approximation technique of [6] and [9] yield C 00 approximations which are without critical points (away from the minimum set). Although the main goal is the investigation of the convex function case, some attempt has been made to separate the arguments which depend explicitly on convexity from those which apply in more general circumstances.…”

“…A typical instance of this situation is in the study of geodesically convex functions on Riemannian manifolds. A procedure is known ( [6], [9]) for approximation of geodesically convex functions by C 00 functions which in a suitable sense are almost convex; and the approximations of strongly convex functions can be taken to be strongly convex. But it remains unknown whether an arbitrary convex function on a Riemannian manifold can be approximated by C 2 convex functions.…”

A note on divergence of $L^{p}$-integrals of subharmonic functions and its applications

“…We may assume that there exists a non-constant continuous subharmonic function v with sup M v ≤ 0. By Corollary 1 in [GW2], for a monotone increasing sequence {r j } of positive numbers tending to infinity, there exists a sequence {v j } of smooth subharmonic functions on M such that sup Bx(rj ) |v − v j | ≤ 1 j for any j ∈ N. Putting u := exp v and u j := exp v j − 1 j , we get 0 < u j ≤ u ≤ 1 on B x (r j ). Here, applying the notation (4) to u = u j and p = 2, we claim that there exists k * ∈ N such that lim…”

A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold

“…Then Na is a closed totally convex subset in M, since pN is a convex function on M. Furthermore Corollary (2.51) implies that the square of the distance pa to Na is strictly convex on M\Na. Therefore by the smooth approximation theorem of Greene and Wu [13], we see that there exists a family {2V}>0 Eof totally convex subsets of M such that (1) the boundary 6NE of N . is smooth for each e>0, (2) limE~0 pE=pa uniformly on each compact set of M\Na (p=disM(N E, *)), and (3) for every compact set U of M'Na, there is a family {o e}>0 Eof real numbers so that limE~0 of=0 and Q11a(pe)<_of on U.…”