Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions

For every $n\geq3$, let $u\in C^\infty(B_2)$ be an admissible solution of $$\kappa[u]\in\Gamma_2,\qquad \sigma_2(\kappa[u])=1,$$ with $$\|u\|_{L^\infty(B_2)}+\|Du\|_{L^\infty(B_2)}\le K.$$ Then $$\sup_{B_{1/2}}|\kappa[u]|\le C(n,K).$$ Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension $n\ge3$. Prior unrestricted quantitative graphical estimates were known in dimension $3$; dimension $4$ had only an implicit estimate with extra dependence, and $n\ge5$ required additional semiconvexity assumptions.

For every $n\geq3$, let $u\in C^\infty(B_2)$ be an admissible solution of $$\kappa[u]\in\Gamma_2,\qquad \sigma_2(\kappa[u])=1,$$ with $$\|u\|_{L^\infty(B_2)}+\|Du\|_{L^\infty(B_2)}\le K.$$ Then $$\sup_{B_{1/2}}|\kappa[u]|\le C(n,K).$$ Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension $n\ge3$. Prior unrestricted quantitative graphical estimates were known in dimension $3$; dimension $4$ had only an implicit estimate with extra dependence, and $n\ge5$ required additional semiconvexity assumptions.