IUMJ

Title: Removable singularities for nonlinear subequations

Authors: H. Lawson and F. Reese Harvey

Issue: Volume 63 (2014), Issue 5, 1525-1552

Abstract:

Let $F$ be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold $X$. We study the question: Which closed subsets $E\subset X$ have the property that every $F$-subharmonic function (subsolution) on $X-E$, which is locally bounded across $E$, extends to an $F$-subharmonic function on $X$? We also study the related question for $F$-harmonic functions (solutions) which are continuous across $E$. The main result asserts that if there exists a convex cone subequation $M$ such that $F+M\subset F$, then any closed set $E$ which is $M$-polar has these properties. $M$-\emph{polar} means that $E=\{\psi=-\infty\}$ where $\psi$ is $M$-subharmonic on $X$ and smooth outside of $E$. Many examples and generalizations are given. These include removable singularity results for \emph{all branches} of the complex and quaternionic Monge-Amp\`ere equations, and a general removable singularity result for the harmonics of geometrically defined subequations. For pure second-order subequations in ${\mathbb {R}}^n$ with monotonicity cone $M$, the \emph{Riesz characteristic} $p=p_M$ is introduced, and extension theorems are proved for any closed singular set $E$ of locally finite Hausdorff $(p-2)$-measure. This applies, for example, to branches of the equation $\sigma_k(D^2 u)=0$ ($k$th elementary function) where $p_M=n/k$, and to its complex and quaternionic counterparts where $p_M = \frac{2n}{k}$, and $p_M=\frac{4n}{k}$, respectively. For convex cone subequations themselves, several removable singularity theorems are proved, independent of the results above.