Optimal concavity of some Hessian operators and the prescribed σ 2 curvature measure problem

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

“…The proof of Theorem 1 follows the same lines of proof Theorem 5 as the isometrically embedded hypersurface obeys the same curvature equation (2). We may use Corollary 2.2 in place of Corollary 2.1 in the proof of Theorem 5.…”

“…Suppose W ∈ Γ 2 is diagonal and W 11 ≥ · · · ≥ W nn , then there exist c 1 > 0 and c 2 > 0 depending only on n such that (7) σ 11 2 (W )W 11 ≥ c 1 σ 2 (W ), and for any j ≥ 2 (8) σ jj 2 (W ) ≥ c 2 σ 1 (W ). The following lemma can be found in [2].…”

“…Then rotate {e 2 , e 3 , · · · e n } such that h ij (x 0 ) is diagonal. Denote F ij = ∂σ 2 ∂h ij , which is positive definite when κ ∈ Γ 2 . At point x 0 ,…”

“…The above result is related to a longstanding problem in fully nonlinear partial differential equations: the interior C 2 estimate for solutions of the following prescribing scalar curvature equation and σ 2 -Hessian equation, (2) σ 2 (κ 1 (x), · · · , κ n (x)) = f (X, ν(x)) > 0, X ∈ B r × R ⊂ R n+1 and (3) σ 2 (∇ 2 u(x)) = f (x, u(x), ∇u(x)) > 0, x ∈ B r ⊂ R n where κ 1 , · · · , κ n are the principal curvatures and ν the normal of the given hypersurface as a gragh over a ball B r ⊂ R n respectively, σ k the k-th elementary symmetric function, 1 ≤ k ≤ n. Equations (2) and (3) are special cases of σ k -Hessian and curvature equations developed by Caffarealli-Nirenberg-Spruck in [1], as an integrated part of fully nonlinear PDE. A C 2 function u is called an admissible solution to equation (3) if u satisfies the equation and ∆u > 0.…”

The interior C2 estimate for the Monge–Ampère equation in dimension n = 2

“…As we all know, it is hard to find a corresponding geometry in higher dimensions, so we can not generalize Heinz's proof or Warren-Yuan's proof to higher dimensions. But the method in this paper and the optimal concavity in [2] is helpful for this problem.…”