Hessian estimates for the sigma‐2 equationin dimension 3

Abstract

We derive a priori interior Hessian estimates for the special Lagrangian equation 2 D 1 in dimension 3.

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

“…For j ≥ 2, σ jj 2 κ j = Ric jj . By Lemma 1, σ jj 2 ≥ cκ 1 , we get (16). Note that σ 3 (κ 1 , · · · , κ n ) = i<j<l κ i κ j κ l .…”

The Monge–Ampère equation for non-integrable almost complex structures

“…We apply the maximum principle to a quantity involving the largest eigenvalue λ 1 (cf. [54,40,41]) of the real Hessian of ϕ. This gives us some good third order terms which are sufficient, after a series of rather technical lemmas, to push the argument through.…”

Singularity and decay estimates for the conformal k-hessian equation via Liouville-type theorems

“…Moreover, some Liouville-type theorems for k-hessian equations were obtained in [4,15] etc. We hope to obtain similar singularity and decay estimates for k-hessian equations, however it (except for k = 2 and dimension 3 [20]) there is presently no local estimate as Lemma 2.1 of conformal k-hessian equation at present.…”