A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotient

Abstract: In this note, we prove a priori interior curvature bounds for strictly convex solutions to elliptic Weingarten curvature quotient equations. The proof does not employ the advanced methods involving integral estimates or compactness arguments. Instead, it relies on a concavity inequality for the equation operator and a novel choice of the auxiliary function to carry out an elementary maximum-principle argument. Interior curvature estimates for strictly convex solutions to the special Lagrangian curvature equation in low dimensions also follow as a consequence.

• The paper extends an elementary proof method for deriving $C^2$ interior curvature estimates so that it applies to the full range of $k$, covering the previously open special cases $3 times k times n-1$.
• The solution involves the adaptation of maximum principles and Bernstien techniques. Particularly,i it stands out for utilizing concavity inequalities established independently by Li-Wu and Dong-Zhang during its formulation.
• The maximum principle and Bernstien techniques used in this innovation do not apply only for the end point usefulness seen in the special cases $k=2$ or $k=n$.

Context and motivation

The paper by Bin Wang (2608.20188) addresses interior $C^2$ estimates for strictly convex solutions of the prescribed curvature quotient equation

$$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$

where $\kappa[\Sigma]$ denotes the principal curvatures of a graph $\Sigma=(x,u(x))$ in $\mathbb{R}^{n+1}$, $\sigma_j$ are elementary symmetric polynomials, and $\psi$ is smooth and positive. The problem of purely local second derivative estimates for Hessian and Weingarten equations has a long history. For the Monge–Ampère equation, Heinz obtained an interior bound in dimension two, while Pogorelov exhibited non-$C^2$ generalized solutions with analytic right-hand side; Urbas extended this negative phenomenon to $\sigma_k$ equations when $3\le k\le n$. The remaining open cases concern the $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$0 equation in dimensions $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$1, and quotient equations $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$2 with $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$3.

For curvature quotients specifically, the case $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$4 admits an elementary pointwise proof due to Sheng–Urbas–Wang, exploiting special algebraic structure and a positive term from the commutation formula for covariant derivatives of the second fundamental form. The quotient $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$5 was treated elementarily by Jianxiang Liu, building on Guan–Qiu's maximum-principle analysis for the $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$6 equation. Wang's contribution extends this to intermediate indices $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$7, where neither convenient algebraic structure nor a distinguished index is available.

Main result

The central theorem states: if $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$8 is smooth on $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$9 with strictly positive principal curvatures and solves $\kappa[\Sigma]$0 for $\kappa[\Sigma]$1, then

$\kappa[\Sigma]$2

with $\kappa[\Sigma]$3 depending on $\kappa[\Sigma]$4, the $\kappa[\Sigma]$5 norm of $\kappa[\Sigma]$6, and the $\kappa[\Sigma]$7 and $\kappa[\Sigma]$8 norms of $\kappa[\Sigma]$9 and $\Sigma=(x,u(x))$0. This covers precisely the previously untreated range $\Sigma=(x,u(x))$1: the endpoint cases $\Sigma=(x,u(x))$2 and $\Sigma=(x,u(x))$3 were already known via Guan–Qiu and Liu respectively.

Two structural remarks qualify the theorem. First, although stated for strictly convex ($\Sigma=(x,u(x))$4) solutions, the proof actually works for strictly $\Sigma=(x,u(x))$5-convex solutions; whether it holds for merely strictly $\Sigma=(x,u(x))$6-convex solutions remains unknown. Second, the proof depends on a concavity inequality for the operator $\Sigma=(x,u(x))$7 on the positive cone, established independently by Li–Wu and Dong–Zhang; an earlier version of the manuscript assumed this inequality as a hypothesis before those preprints appeared.

As a consequence, since $\Sigma=(x,u(x))$8 coincides with the special Lagrangian curvature equation $\Sigma=(x,u(x))$9 in dimensions $\mathbb{R}^{n+1}$0 — via the tangent angle-sum identity, whose numerator reduces to $\mathbb{R}^{n+1}$1 — the paper obtains interior curvature estimates for strictly convex solutions to that equation with variable right-hand side. When $\mathbb{R}^{n+1}$2, Qiu–Zhou had proved such estimates in all dimensions using integral methods; the present argument provides an elementary alternative in low dimensions.

Method of proof

The proof deliberately avoids integral estimates and compactness arguments, relying instead on a Bernstein-type maximum principle applied to the auxiliary function

$\mathbb{R}^{n+1}$3

which is exactly the auxiliary function introduced by Guan–Qiu for the $\mathbb{R}^{n+1}$4 equation. The argument proceeds in two parts.

Part I derives a Jacobi inequality for $\mathbb{R}^{n+1}$5 (or a smooth majorant $\mathbb{R}^{n+1}$6 when $\mathbb{R}^{n+1}$7 has multiplicity $\mathbb{R}^{n+1}$8, handled via the Brendle–Choi–Daskalopoulos approximation lemma). Contracting second covariant derivatives of $\mathbb{R}^{n+1}$9 against $\sigma_j$0, where $\sigma_j$1, and expanding through the commutator formula, Codazzi equation, Gauss equation, and double differentiation of the PDE, the key step applies the Li–Wu concavity inequality to the vector $\sigma_j$2, whose first $\sigma_j$3 components coincide by the first-order approximation relations. The remaining third-order terms combine non-negatively because all curvatures are positive, yielding

$\sigma_j$4

A supporting lemma shows that for $\sigma_j$5, $\sigma_j$6 dominates a fixed fraction of $\sigma_j$7, while for $\sigma_j$8 it satisfies $\sigma_j$9 — the latter requiring $\psi$0 and Newton–Maclaurin type inequalities.

Part II carries out the maximum-principle analysis at the interior maximum of $\psi$1, split into three cases according to the tangential position vector. In Case 1 ($\psi$2), choosing $\psi$3 large absorbs the problematic terms. In Case 2 (the dominant index $\psi$4), either $\psi$5 is small, in which case the gradient of $\psi$6 in direction $\psi$7 is large enough to dominate via $\psi$8, or $\psi$9, in which case $C^2$0 bounds $C^2$1 directly. Case 3 ($C^2$2) is the technically delicate part: since no single distinguished index exists for general $C^2$3, one must show there is some $C^2$4 with $C^2$5 bounded below in terms of $C^2$6. This claim is proved by expressing each $C^2$7 linearly in the $C^2$8, splitting indices into $C^2$9 and $\sigma_k$0, and combining Cauchy–Schwarz estimates with the constraint that $\sigma_k$1 is small whenever Case 2 fails. If instead $\sigma_k$2, the estimate follows from $\sigma_k$3.

The dependence of the constant on $\sigma_k$4 enters through uniform lower bounds on $\sigma_k$5 and comparability estimates for the $\sigma_k$6; gradient-dependent right-hand sides would break the argument, consistent with the known failure of $\sigma_k$7 estimates for quotient equations with gradient terms noted by Guan–Ren–Wang.

Limitations and open questions

Three limitations are explicit in the paper. First, the theorem requires strict convexity ($\sigma_k$8); the extension to strictly $\sigma_k$9-convex solutions uses the Dong–Zhang concavity inequality together with a lower bound $3\le k\le n$0, but the strictly $3\le k\le n$1-convex case is left unresolved. Second, the method depends on the Li–Wu/Dong–Zhang concavity inequality for $3\le k\le n$2 on the positive cone; without it the Jacobi inequality cannot be closed. Third, whether the integral approach of Lu–Tsai transfers to curvature quotient equations is untested — the author states plainly that no attempt has been made, so the present theorem does not follow automatically from the Hessian-side results. More broadly, the ultimate goal of interior estimates for strictly $3\le k\le n$3-convex solutions to $3\le k\le n$4 remains open, as does the analogous question for the $3\le k\le n$5 equation in dimensions $3\le k\le n$6.

Conclusion

This note completes the elementary pointwise treatment of interior curvature estimates for the curvature quotient family $3\le k\le n$7 across all $3\le k\le n$8, extending Guan–Qiu's auxiliary-function technique beyond the structurally favorable endpoints $3\le k\le n$9 and $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$00. The main technical innovation is the index-splitting argument establishing the lower bound on $$\left(\frac{\sigma_k}{\sigma_{k-2}}\right)(\kappa[\Sigma])=\psi(X),$$01 in Case 3, which substitutes for the single distinguished index available in the endpoint cases. Combined with the recent concavity inequalities of Li–Wu and Dong–Zhang, the result also yields elementary proofs of interior estimates for the special Lagrangian curvature equation in dimensions three and four with variable phase function.