Removable Singularities for Nonlinear Subequations

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 ⊂ 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 ⊂ F , then any closed set E which is M -polar has these properties. M -polar means that E = {ψ = −∞} where ψ is M -subharmonic on X and smooth outside of E. Many examples and generalizations are given. These include removable singularity results for all branches of the complex and quaternionic Monge-Ampère equations, and a general removable singularity result for the harmonics of geometrically defined subequations. For pure second-order subequations in Rn with monotonicity cone M the Riesz characteristic p = pM 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 σk(D 2u) = 0 (kth elementary function) where pM = n/k, and its complex and quaternionic counterparts where pM = 2n k and pM = 4n k respectively. For convex cone subequations themselves, several removable singularity theorems are proved, independent of the results above. 1 Partially supported by the N.S.F. Date: March 5, 2014. 1