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 Fsubharmonic 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 Mpolar has these properties. M-polar means that E = { ψ = -∞} where 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 = 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 σ_k(D²u) = 0 (kth elementary function) where pM = n/k, and to its complex and quaternionic counterparts where p_M = 2n/k , and p_M = 4n/k , respectively. For convex cone subequations themselves, several removable singularity theorems are proved, independent of the results above.