Every smoothly bounded p-convex domain in R^n admits a p-plurisubharmonic defining function

TL;DR

The paper proves that smoothly bounded p-convex domains in R^n have p-plurisubharmonic defining functions, and explores implications for conformal harmonic maps, including extension properties.

Contribution

It establishes the existence of p-plurisubharmonic defining functions for p-convex domains and analyzes their impact on harmonic maps and boundary behavior.

Findings

Existence of smooth p-plurisubharmonic defining functions for p-convex domains.

Extension of conformal harmonic maps from punctured disks to the full disk.

Characterization of harmonic maps with boundary in p-convex domains.

Abstract

We show that every bounded domain $D$ in $\mathbb R^n$ with smooth $p$-convex boundary for $2\le p < n$ admits a smooth defining function $\rho$ which is $p$-plurisubharmonic on $\overline D$; if in addition $bD$ has no $p$-flat points then $\rho$ can be chosen strongly $p$-plurisubharmonic on $D$. If $bD$ is $2$-convex then for any open connected conformal surface $M$ and conformal harmonic map $f:M\to \overline D$, either $f(M)\subset D$ or $f(M)\subset bD$. In particular, every conformal harmonic map $\mathbb D^*\to D$ from the punctured disc extends to a conformal harmonic map $\mathbb D\to D$.