Paper detail

Tangents to subsolutions -- existence and uniqueness, Part I

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the &#34;Riesz characteristic&#34;, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Ampère equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $\geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Hölder continuity of subsolutions when the Riesz characteristic p satisfies $1 \leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the &#34;geometric cases&#34;. A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Ampère case are completely classified.