The graphical scalar curvature equation prescribes the scalar curvature of a hypersurface given as the graph of a function 
on a domain in 
. If 
, 
, ..., 
are its principal
curvatures, the normalization of constant scalar
curvature considered here is
 |
With the convention 
, the scalar curvature
is 2. The admissible elliptic branch is specified by 
and 
, where 
. The equation is a fully nonlinear second-order
partial differential equation for
.
Qiu and Yan (2026) proved interior curvature estimates in all dimensions 
. For a smooth admissible solution on the radius-2 ball
with a fixed bound 
on its height and gradient, the
absolute values of its principal curvatures
on the concentric radius-
ball are bounded by a constant depending only on 
and 
. Thus the interior second-order control does not require a
boundary curvature bound.
The authors disclosed AI assistance with exploratory arguments and computations and took responsibility for checking and correcting the proof. Independent external review had not been reported as of Sep. 7, 2026.
