Smooth approximation of plurisubharmonic functions on almost complex manifolds

Abstract

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X, J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence $u_j \in C^\infty (X)$ of smooth strictly J-plurisubharmonic functions point-wise decreasing down to u. In any almost complex manifold (X, J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.

Appendices

Appendix 1: Affine jet-equivalence

A local affine jet-equivalence is a local isomorphism of the (canonically trivialized) 2-jet bundle $\mathbf{J}({\mathbb R}^n) = {\mathbb R}\times {\mathbb R}^n\times {\mathrm{Sym}^2({\mathbb R}^n)}$ which is of the form:

$$\begin{aligned} r' = r+r_0(x), \quad p'=k(x)p + p_0(x),\quad A'= h(x) A h(x)^t + L_x(p) +A_0(x) \end{aligned}$$

where

$$\begin{aligned}&r_0(x) \ \text {takes values in}\quad {\mathbb R}, \\&p_0(x) \ \text {takes values in}\quad {\mathbb R}^n, \\&A_0(x) \ \text {takes values in}\quad {\mathrm{Sym}^2({\mathbb R}^n)}, \\&\text {(i.e.,}\quad J_0(x) \equiv (r_0(x), p_0(x), A_0(x)) \mathrm{\quad is\,a\, section\,of} \mathbf{J}({\mathbb R}^n)) \end{aligned}$$

and

$$\begin{aligned}&k(x)\,\mathrm{and}\,h(x) \mathrm{\quad take\,values\,in}\,\mathrm{GL}_n({\mathbb R}), \mathrm{while} \\&L_x\,\mathrm{takes\,values\,in}\,\mathrm{Hom\,}({\mathbb R}^n, {\mathrm{Sym}^2({\mathbb R}^n)})\\ \end{aligned}$$

The regularity conditions on the jet-equivalence required in the proof of Theorem 10.1 in [10] are:

• (1) k, h and L are Lipschitz continuous, and

• (2) $J_0$ is continuous.

For the second jet equivalence in our application to the Obstacle Problem, $g\equiv h \equiv Id$ and $J_0(x) = (r_0(x),0,0)$, so our obstacle function $g(x) = -r_0(x)$ need only be continuous.

Appendix 2: $\Sigma _m$-Subharmonic functions

As noted in Remark 1.3, for any subequation F, smooth approximation for F-subharmonic functions can be proved whenever continuous approximation and a Richberg-type theorem can be established for F. In this appendix we give just such a result for the complex hessian subequations on a Kähler manifold.

Let X be a complex manifold of dimension n with a fixed Kähler form $\omega$. We say that a function $u\in \mathcal {C}^2(\Omega )$ is $\Sigma _m$-subharmonic on a domain $\Omega \subset \subset X$ if $(dd^cu)^k\wedge \omega ^{n-k}\ge 0$ for $k=1,\ldots ,m$. We say that a upper semi-continuous function

$$\begin{aligned} u:\Omega \rightarrow [-\infty ,+\infty ) \end{aligned}$$

is $\Sigma _m$-subharmonic ($u\in {\Sigma }_m(\Omega )$) if u is locally integrable and

$$\begin{aligned} dd^cu\wedge dd^cu_1\wedge \cdots \wedge dd^cu_{m-1}\wedge \omega ^{n-m}\ge 0, \end{aligned}$$

for any $\mathcal {C}^2$ $\Sigma _m$-subharmonic functions $u_1,\ldots u_{m-1}$ (they are defined in [1] for $\omega =\omega _{st}=dd^c(|z|^2)$ in $\mathbb {C}^n$ and in [5, 14] for general Kähler form). It is non-trivial that this distributional definition coincides with the viscosity definition (see [12, App. A] and forward references). This is just the subequation $F\equiv \Sigma _m$ defined on X by the condition that the first m elementary symmetric functions of the complex hessian satisfy $\sigma _\ell (\mathrm{Hess}_{\mathbb C}u) \ge 0$ for $\ell =1,\ldots ,m$ (compare Example 18.1 in [10] and Lemma 7 in [20]).

A Richberg-type theorem for ${\Sigma }_m$ was proved in [20] (Theorem 2). Lu and Nguyen proved in [15] that on compact Kähler manifolds any quasi-$\Sigma _m$-subharmonic function can be approximated from above by smooth quasi-$\Sigma _m$-subharmonic functions (a function u is quasi-$\Sigma _m$-subharmonic if the function $u+\rho$ is $\Sigma _m$-subharmonic where $\rho$ is local potential for $\omega$). Actually their global result implies that locally it is possible to regularize $\Sigma _m$-subharmonic functions. However, in the same way as in Theorem 4.1, we can prove a slightly stronger result.

Theorem 6.1

Let X be a $\Sigma _m$ -pseudoconvex Kähler manifold. Let u be a $\Sigma _m$ -subharmonic function on X. Then there exists a decreasing sequence $u_j \in \mathcal {C}^\infty (X)$ of $\Sigma _m$ -subharmonic functions such that $u_j\downarrow u$.

By $\Sigma _m$ -pseudoconvex we mean that X has a global $\mathcal {C}^2$ strictly $\Sigma _m$-subharmonic exhaustion function. In particular Stein manifolds are $\Sigma _m$-pseudoconvex.