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A concavity inequality and interior $C^2$ estimate for Hessian quotient equations

This paper establishes a new concavity inequality and Jacobi inequality for Hessian quotient operators $\sigma_k/\sigma_l$ (where $k-l \in \{1,2\}$ ), which are used to derive interior $C^2$ estimates for convex solutions and prove that entire convex solutions with quadratic growth must be quadratic polynomials.

Original authors: Zhisu Li, Ke Wu

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Zhisu Li, Ke Wu

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding how shapes curve and change, particularly when those shapes are defined by complex equations rather than simple lines. Imagine a surface that can bend in many different directions at once, like a crumpled piece of paper that has been smoothed out but still holds the memory of its folds. Mathematicians study these surfaces to understand the fundamental rules that govern their behavior. A central question in this field is whether a surface that looks smooth and well-behaved on the inside must also be smooth everywhere, or if it can suddenly develop sharp, jagged points that break the rules of smoothness. For decades, researchers have known that in certain dimensions, these surfaces can indeed develop such singularities, or sharp breaks, unless specific conditions are met. The challenge has been to find the precise conditions that guarantee a surface remains smooth, especially when the equations describing them involve a mix of different types of curvature.

A team of mathematicians, Zhisu Li and Ke Wu, has now solved a long-standing puzzle regarding a specific family of these complex equations. They focused on a scenario where the curvature is defined by a ratio of two different measures of bending. Think of it as comparing the total amount of bending in one direction against the bending in another. The researchers were interested in cases where the difference between these two measures is very small—specifically, when the gap is just one or two steps in a mathematical sequence. In these specific cases, previous work had left a gap in understanding: it was not known whether the smoothness of the surface could be guaranteed without extra, artificial assumptions. Li and Wu proved that for these specific ratios, the smoothness is indeed guaranteed. They showed that if a solution to these equations is convex—meaning it curves outward like a bowl rather than inward like a saddle—and grows at a steady, quadratic rate, then the surface cannot have any hidden sharp points. It must be perfectly smooth throughout.

The core of their discovery lies in a new mathematical inequality, a rule that acts like a safety net for the curvature. To find this rule, the authors had to look closely at the behavior of the largest curves on the surface. They demonstrated that when the largest curves are equal in size, a specific relationship holds between the different ways the surface bends. This relationship is strong enough to prevent the curves from becoming infinitely sharp, which is what would happen if a singularity were to form. By establishing this inequality, they were able to prove a "Jacobi inequality," a powerful tool that allows mathematicians to track how the curvature evolves. Using this tool, they followed a path laid out by previous researchers to show that the curvature remains bounded and controlled. This means the surface cannot suddenly spike or break; it stays within a predictable range of smoothness.

The implications of this finding extend beyond just proving smoothness. The researchers also applied their result to a famous type of problem known as a Liouville theorem, which asks whether a solution that exists everywhere in space and grows in a specific way must be a simple, predictable shape. In this case, they considered solutions that grow like a parabola, getting larger as you move away from the center. They proved that any such solution must be a simple quadratic polynomial. In plain terms, this means the surface is not a complicated, twisted shape at all, but a perfect, smooth bowl or dome. This result completes the classification of these equations for convex solutions, filling in the final missing pieces of the puzzle for these specific ratios.

Before this work, there were known examples where similar equations produced jagged, singular solutions, but those examples required a larger gap between the two measures of curvature. The new proof shows that when the gap is small, nature does not allow those jagged breaks to happen. The authors did not just suggest this might be true; they provided a rigorous, step-by-step proof that leaves no room for doubt. They also showed that their method works for a wide range of dimensions, not just in two or three dimensions, but in any number of dimensions. This universality is significant because it suggests a deep, underlying order in how these complex shapes behave.

The journey to this result involved overcoming a significant algebraic hurdle. The researchers had to prove that a certain inequality holds true for a very specific set of directions on the surface, rather than for every possible direction. This was a clever simplification that allowed them to bypass previous limitations. By focusing on the directions where the largest curves are equal, they could derive the necessary bounds without needing the stronger, more restrictive conditions that were previously thought to be required. This insight allowed them to extend the known results to cover the remaining cases that had stumped mathematicians for years.

Ultimately, this paper provides a definitive answer to a question that has been open for some time. It confirms that for a specific class of curvature equations, smoothness is an inherent property of convex solutions that grow at a quadratic rate. There are no hidden singularities waiting to be discovered in these cases. The surface is guaranteed to be a simple, smooth quadratic shape. This finding brings a sense of closure to a chapter in the theory of nonlinear equations, offering a clear picture of how these complex shapes behave when the conditions are just right. It stands as a testament to the power of precise mathematical reasoning to reveal the hidden order in the most intricate geometric forms.

Technical Summary: A Concavity Inequality and Interior $C^2$ Estimate for Hessian Quotient Equations

Problem Statement
The paper addresses the fundamental problem of establishing interior Hessian estimates (interior $C^2$ regularity) for convex solutions of fully nonlinear elliptic Hessian quotient equations of the form:
$F(D^2u) := \frac{\sigma_k(D^2u)}{\sigma_l(D^2u)} = f(x, u)$
where $1 \le l < k \le n$ and $\sigma_k$ denotes the $k$ -th elementary symmetric function of the eigenvalues of the Hessian matrix $D^2u$ .

While interior $C^2$ estimates are well-understood for the Monge-Ampère equation ( $k=n, l=0$ ) in dimension $n=2$ , and for specific cases like $\sigma_2(D^2u)=f$ under various structural assumptions, significant gaps remain for general Hessian quotients. Previous work by Pogorelov and Urbas demonstrated that such estimates fail for $k \ge 3$ without additional structure. Recent progress by Lu and Tsai [LT26] established interior estimates for adjacent quotients ( $k-l=1$ ) and the case $k-l=2$ , but their framework relied on an algebraic assumption (Assumption 1.1 in [LT26]) regarding a concavity inequality for the operator $F$ . This paper aims to rigorously prove that assumption for the cases $k-l \in \{1, 2\}$ .

Methodology
The authors employ a multi-step analytical approach combining algebraic inequalities, geometric transformations, and regularity theory:

Algebraic Concavity Inequality: The core of the work is the proof of a quantitative concavity inequality for the Hessian quotient operators. The authors focus on the specific scenario where the largest eigenvalue $\lambda_{\max}$ has multiplicity $m$ . They restrict their analysis to directions $\xi$ where the first $m$ components are equal ( $\xi_1 = \dots = \xi_m$ ), which corresponds to the derivatives arising in the Jacobi inequality calculation for the largest eigenvalue.
- For the case $k-l=1$ (adjacent quotient $\sigma_k/\sigma_{k-1}$ ), they establish the inequality directly using properties of the quotient $q_k = \sigma_k/\sigma_{k-1}$ .
- For the case $k-l=2$ , they utilize a "one-dimensional lifting" technique. They embed the eigenvalues $\lambda \in \mathbb{R}^n$ into $\mathbb{R}^{n+1}$ via $\Lambda = (\lambda, \sqrt{F(\lambda)})$ . This transformation reduces the problem to an adjacent quotient in a higher dimension, allowing the application of the $k-l=1$ results.
- The proof of the key inequality (Lemma 2.4) utilizes a contradiction argument involving asymptotic analysis of the eigenvalues and the use of Newton inequalities.
Jacobi Inequality and Legendre Transform: Once the concavity inequality is established, the authors derive a Jacobi inequality for $b = \log \lambda_{\max}(D^2u)$ . Following the framework of Lu and Tsai [LT26], they combine this Jacobi inequality with the Legendre transform, the mean value inequality, and integration by parts to derive the interior Hessian estimate.
Liouville Theorem: To prove the rigidity of entire solutions, the authors apply the constant-rank theorem of Bian and Guan to show that the Hessian of a convex entire solution has full rank. They then use the Legendre transform to map the problem to a dual equation. By establishing uniform bounds on the dual solution and applying Evans-Krylov estimates, they demonstrate that the Hessian of the dual solution is constant, implying the original solution is a quadratic polynomial.

Key Contributions and Results

Theorem 1.1 (Concavity Inequality): The authors prove that for $k-l \in \{1, 2\}$ , there exist constants $A$ and $K$ such that a specific concavity inequality holds for the Hessian quotient operator $F$ when the largest eigenvalue is sufficiently large and the test vector $\xi$ satisfies the multiplicity condition ( $\xi_1 = \dots = \xi_m$ ). This result validates the structural assumption required by the Lu-Tsai framework.
Theorem 1.2 (Interior $C^2$ Estimate): As a direct consequence of Theorem 1.1 and the Lu-Tsai framework, the paper establishes an interior Hessian estimate for convex solutions of $\sigma_k(D^2u)/\sigma_l(D^2u) = f$ in the cases $k-l \in \{1, 2\}$ . Specifically, $|D^2u(0)| \le C$ , where $C$ depends on dimension, the solution's $C^{0,1}$ norm, and the regularity of $f$ .
Theorem 1.3 (Liouville Theorem): The paper proves that any entire convex solution $u \in C^4(\mathbb{R}^n)$ to $\sigma_k(D^2u)/\sigma_l(D^2u) = 1$ with quadratic growth ( $u(x) \ge a|x|^2 - c$ ) must be a quadratic polynomial. This extends previous rigidity results to the cases $k-l \in \{1, 2\}$ .

Significance and Claims
The paper claims to complete the classification of interior $C^2$ regularity for convex solutions of Hessian quotient equations. By resolving the algebraic concavity inequality for the cases $k-l=1$ and $k-l=2$ , the authors fill the remaining gaps in the theory where singular solutions are not known to exist (unlike the cases $l \le k-3$ ).

The authors note that their result, combined with known singular examples for $l \le k-3$ , provides a sharp dichotomy: interior $C^2$ estimates hold for convex solutions if and only if $k-l \in \{1, 2\}$ (under the structural conditions addressed). Furthermore, the Liouville theorem extends the rigidity phenomenon observed in specific quotient equations to this broader class, confirming that quadratic growth forces the solution to be a quadratic polynomial.

The work relies on the framework developed by Lu and Tsai [LT26] and extends it by providing the necessary algebraic foundation. The authors also acknowledge that Tsai [T26] independently proved the assumption for the $k-l=1$ case using a different method (change of basis for symmetric polynomials), but this paper provides a unified approach for both $k-l=1$ and $k-l=2$ using the lifting technique for the latter.

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