A bound inside the smaller ball
A mathematical preprint reports an interior upper bound for the largest principal curvature of smooth, strictly convex graph solutions. Principal curvature is a measure of how a graph bends in a particular direction; the result says that, inside the ball with half the original radius, the absolute value of the largest such curvature is no greater than a constant C. The numerical value of C is not specified. Instead, the bound allows C to depend on n, k and r, the graph’s C1 norm, and the C2 and L∞ norms of the positive right-hand side ψ together with the L∞ norm of 1/ψ—mathematical measures of the size and variation of the graph and the prescribed function.
The theorem’s setting is narrow but explicit. It considers 2 ≤ k ≤ n, a smooth positive ψ, and smooth strictly convex graph solutions whose principal curvatures are positive in the larger ball B_r. In other words, the bound applies only after the solution has met the note’s stated analytic and geometric conditions; it is not presented as a result for arbitrary graphs.
A pointwise proof
The note’s stated objective is to extend Liu’s result to a more general prescribed curvature quotient problem. In this type of equation, a quotient built from curvature quantities is set equal to a prescribed right-hand side, here represented by ψ. The highlighted extension allows that right-hand side to be non-constant and covers all 3 ≤ k ≤ n − 1. The theorem statement itself is framed for 2 ≤ k ≤ n, making the scope and the assumptions different parts of the result.
To prove the estimate, the author avoids integral estimates and compactness arguments. The route instead combines a concavity inequality, a Jacobi inequality and a novel auxiliary function with an elementary maximum-principle argument. The maximum principle lets the calculation use the point where the auxiliary quantity reaches its peak to control what happens across the interior.
At that maximum point, the auxiliary quantity b satisfies a Jacobi-type inequality containing a positive gradient-square term together with curvature and bounded-error terms. This is the key local inequality used in the maximum-principle calculation.
A special-Lagrangian consequence
The note also gives a corollary for the special-Lagrangian curvature equation in dimensions n = 3 and n = 4, assuming a smooth positive ψ. A low-dimensional angle calculation turns that equation into the relation σ3/σ1 = [ψ(x, u(x))]^2 in B_r. The relation is the bridge the note uses to place this named geometric equation within the broader quotient framework.
In that setting, the claimed conclusion has the same interior form: the maximum absolute principal curvature is bounded by a constant C on B_{r/2}. The constant is allowed to depend on n, r, the graph norm and the stated norms of ψ, but the corollary does not supply a numerical value.
Where the claim stops
The result has a defined boundary. The proof is stated to remain valid for strictly (k + 1)-convex solutions, but validity for strictly k-convex solutions is explicitly unknown. The note therefore treats the strictly k-convex case as an open problem, not as an established extension.
The document is presented as an arXiv preprint. Its acknowledgments report support from the China Postdoctoral Science Foundation under Grant Number 2026M793406.
Paper data and sources
Original title: A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotient
Authors: Bin Wang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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