Positive Hessian Quotient Equation

• Positive Hessian quotient equations are fully nonlinear elliptic PDEs defined using elementary symmetric functions of the Hessian eigenvalues, with applications in curvature and geometric analysis.
• The theory employs techniques like barrier constructions, Perron’s method, and Jacobi-type inequalities to derive interior estimates, Liouville rigidity, and Bernstein phenomena under various convexity conditions.
• The equation unifies classical cases such as the k-Hessian and Monge–Ampère models, extending its relevance to manifold settings and problems in both boundary and exterior domains.

The positive Hessian quotient equation is a fully nonlinear elliptic partial differential equation built from elementary symmetric functions of the eigenvalues of the Hessian. For a $C^{2}$ function $u$, with Hessian eigenvalues $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$, one sets $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$, where $\sigma_{j}$ is the $j$-th elementary symmetric polynomial, and studies equations of the form

$$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$

In the constant-right-hand-side model one often writes $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$. The subject connects the $k$-Hessian equation, the Monge–Ampère equation, special Lagrangian-type equations, curvature problems, and geometric PDE on Kähler, HKT, Riemannian, and spherical settings. Recent work has developed a sharply differentiated theory: strong Liouville and Bernstein rigidity under integral or semi-convex hypotheses, interior $C^{2}$ estimates in several regimes, explicit counterexamples in others, and boundary/exterior solvability via barrier constructions and Perron’s method (Du, 2021).

1. Operator, admissibility, and basic models

Let $u$0, $u$1, and let $u$2 denote the eigenvalues of $u$3. For $u$4,

$u$5

The basic Hessian-quotient operator is

$u$6

Several papers also use the normalized operator

$u$7

especially in parabolic and boundary-value formulations (Chen et al., 2020).

Ellipticity is tied to the Gårding cone

$u$8

A function is called $u$9-admissible or $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$0-convex when $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$1 at every point. On $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$2, $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$3 for $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$4, so the quotient is well defined and elliptic. In some works, especially those centered on “positive” operators, one imposes the stronger positive cone $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$5 (Sroka, 6 Jan 2025).

The standard special cases organize much of the theory. When $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$6, one recovers the $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$7-Hessian equation $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$8. When $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$9, one obtains the Monge–Ampère equation $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$0. When $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$1, the equation is identified in the supplied literature with the special Lagrangian / Lagrangian angle equation (Du, 2021). In dimension two, $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$2 gives

$\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$3

on a Riemannian surface, a real analogue of a quotient equation that has been compared with the complex $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$4-equation (Sroka, 6 Jan 2025).

The same quotient structure appears in several extensions. One replaces $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$5 by $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$6 on Kähler manifolds (Chen, 2021), by $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$7 on HKT manifolds (Chen, 2022), by a symmetric tensor $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$8 on a Riemannian manifold (Guan et al., 19 Sep 2025), or by the support-function matrix $\sigma_{j}(D^{2}u)=\sigma_{j}(\lambda(D^{2}u))$9 on $\sigma_{j}$0 in Christoffel–Minkowski-type problems (Luo et al., 13 Apr 2026). There are also vector-valued variants based on $\sigma_{j}$1, formed from sums of selected eigenvalues, leading to quotient equations of Hessian-quotient type with gradient-dependent right-hand side (Gong et al., 10 Jan 2025).

2. Entire-solution rigidity and Bernstein phenomena

A central global question is the Bernstein property: whether an entire admissible solution must be a quadratic polynomial. For the quotient equation

$\sigma_{j}$2

an entire convex $\sigma_{j}$3 solution is said to have the Bernstein property if one can prove that $\sigma_{j}$4 is quadratic. Earlier results, including those of Jörgens, Calabi, Pogorelov, Cheng–Yau, and Bao–Cheng–Guan–Ji, imposed pointwise quadratic-growth or uniform convexity assumptions such as $\sigma_{j}$5. Under such assumptions one derives $\sigma_{j}$6 and then uses Liouville-type arguments to force $\sigma_{j}$7 (Du, 2021).

A major advance is the equivalence theorem of “Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation” (Du, 2021). For a locally strictly convex solution normalized by $\sigma_{j}$8, $\sigma_{j}$9, with sublevel sets $j$0, the following are equivalent, and each is necessary and sufficient for $j$1 to be a quadratic polynomial:

$j$2

a reverse isoperimetric-type inequality;

$j$3

a volume-growth condition with sharp exponent $j$4;

and

$j$5

for some $j$6, an $j$7-integrability condition. Under any one of these, the solution has the form

$j$8

This removes the pointwise quadratic-growth assumptions used in earlier work and replaces them with equivalent integral or volume criteria.

For the specific positive quotient $j$9, Lu–Sroka prove a Liouville theorem for admissible semi-convex entire solutions: if $$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$0 satisfies

$$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$1

then $$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$2 is quadratic (Lu et al., 16 Feb 2026). Their method is unusually direct: after the shift

$$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$3

one obtains

$$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$4

and the problem reduces to the Shankar–Yuan Liouville theorem for the pure $$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$5 equation.

Mei–Yan establish analogous rigidity for

$$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$6

in arbitrary dimension, assuming admissibility, semi-convexity, and sub-quadratic growth $$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$7. Under these hypotheses the entire solution is again a quadratic polynomial (Mei et al., 25 Apr 2026). The same paper derives rigidity for the sum-Hessian equation $$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$8 under several alternative lower-Hessian conditions.

A parabolic analogue was established by Dai–Bao–Wang for

$$\frac{\sigma_{k}(D^{2}u)}{\sigma_{\ell}(D^{2}u)}=f, \qquad 0\le \ell<k\le n.$$9

If $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$0 is parabolically convex, satisfies $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$1, and obeys $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$2, then

$\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$3

where $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$4 is constant and $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$5 is a convex quadratic polynomial (Dai et al., 2023).

A frequent misconception is that pointwise quadratic growth is indispensable for Bernstein-type rigidity. The equivalence theorem in (Du, 2021) shows that, for the positive quotient equation, integral and volume conditions can be both weaker and logically complete. Another recurrent misconception is that all Liouville proofs require elaborate global maximum principles; in the $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$6 case, the reduction to a pure $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$7 equation shows otherwise (Lu et al., 16 Feb 2026).

3. Interior regularity and second-order estimates

Interior $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$8 theory for positive Hessian quotient equations is highly regime dependent. Some operators admit sharp local Hessian bounds under natural convexity or semi-convexity hypotheses, while other quotient families exhibit explicit singular solutions, so no general interior $\sigma_{k}(D^{2}u)/\sigma_{\ell}(D^{2}u)=1$9 estimate can hold.

Regime	Assumptions	Result
$k$0 in $k$1	convex $k$2, $k$3, $k$4	interior $k$5 estimate (Lu, 2023)
$k$6	$k$7: 2-convex; $k$8: 2-convex and semi-convex	interior Hessian estimates (Jiao et al., 15 Feb 2026)
$k$9	$C^{2}$0	interior $C^{2}$1 estimate (Lu, 2024)
$C^{2}$2	$C^{2}$3	interior $C^{2}$4 estimate fails (Lu, 2024)
$C^{2}$5, $C^{2}$6	$C^{2}$7-admissible, semi-convex	interior $C^{2}$8 estimate in arbitrary dimensions (Mei et al., 25 Apr 2026)
$C^{2}$9 in $u$00	$u$01, $u$02	interior $u$03 via $u$04 bound (Zhou, 2023)

For $u$05 in dimension three, Lu proves that if $u$06 is convex and solves

$u$07

then

$u$08

The core innovation is a Jacobi-type inequality for $u$09, the logarithm of the largest eigenvalue: $u$10 in the viscosity sense when $u$11 is large (Lu, 2023). The proof combines spectral formulas, a Legendre transform of $u$12, a mean-value inequality, and a divergence-free tensor

$u$13

Jiao–Sui obtain interior Hessian estimates for the quotient $u$14 with right-hand side $u$15. In dimension three they require only 2-convexity, whereas for $u$16 they assume 2-convexity and $u$17-semi-convexity. Their method is based on a doubling inequality

$u$18

a new Lagrange-multiplier argument for third-derivative terms, and a blow-up/compactness argument using Savin’s small-perturbation $u$19 theorem or the Chaudhuri–Trudinger extension (Jiao et al., 15 Feb 2026). In $u$20, the same scheme uses a dynamic semi-convexity ratio bound at the critical point,

$u$21

For the family $u$22, Guan–Sroka’s concavity identities yield a complete dichotomy. If $u$23 or $u$24, there is an interior $u$25 estimate for convex admissible $u$26, with constants depending on interior distance, $u$27-norm of $u$28, $u$29, and $u$30 or $u$31-data. If $u$32, the estimate fails: there exists a convex viscosity solution, Lipschitz but not $u$33 for any $u$34, obtained from an adaptation of Pogorelov’s example (Lu, 2024). This is one of the sharpest negative results in the area and shows that admissibility alone does not secure second-order regularity.

Mei–Yan prove that the equations

$u$35

admit interior $u$36 estimates in arbitrary dimension under the natural hypotheses $u$37 and $u$38. Their proof follows a three-step scheme: a Jacobi inequality for $u$39, a Legendre–Lewy duality argument that produces a uniformly elliptic dual operator, and integration by parts using weight comparisons involving $u$40 and $u$41 (Mei et al., 25 Apr 2026).

A complementary reformulation is due to Zhou. For the twisted special Lagrangian equation

$u$42

he derives a priori $u$43 estimates under only Lipschitz $u$44. In dimension three this implies interior $u$45 regularity for continuous viscosity solutions of

$u$46

with $u$47, $u$48 (Zhou, 2023). This suggests that structural reformulation can substitute for direct quotient estimates in selected low-dimensional settings.

4. Boundary value, exterior, and evolution problems

The positive Hessian quotient equation has a substantial boundary-value theory. In bounded domains, one studies Dirichlet or Neumann problems; in exterior domains, one prescribes asymptotic behavior near infinity; in parabolic settings, one studies long-time convergence and translating solutions.

For exterior domains, Li–Li–Zhao treat

$u$49

with $u$50 smooth, bounded, and strictly convex, boundary data $u$51, and prescribed quadratic asymptotic behavior at infinity (Li et al., 2020). Their construction introduces generalized symmetric subsolutions of the form

$u$52

reducing the PDE to an ODE via identities for $u$53 under rank-one perturbations. Perron’s method then yields a unique $u$54-convex viscosity solution asymptotic to

$u$55

with $u$56 satisfying $u$57. The same paper states solvability for any $u$58, with the abstract emphasizing all dimensions $u$59, while the detailed theorem in the supplied text is stated for $u$60. This suggests that the dimensional range is formulation sensitive in the presentation.

Jiang–Li–Li extend the exterior problem to nonconstant right-hand side

$u$61

again on $u$62, with generalized symmetric asymptotic behavior

$u$63

and $u$64 (Jiang et al., 2022). The proof combines comparison principles on unbounded domains, local quadratic boundary barriers, and explicit generalized symmetric sub- and supersolutions obtained from an ODE for $u$65, $u$66.

The earlier exterior Dirichlet theory of (Li et al., 2017) is organized around new quantities governing generalized radial subsolutions and the asymptotic decay exponent $u$67. There the conclusion is existence and uniqueness of a viscosity solution with prescribed quadratic asymptotics for the constant equation $u$68, again by a Perron construction built on suitable subsolutions.

For Neumann problems, Chen–Ma–Zhang study the parabolic flow

$u$69

with Neumann boundary condition $u$70, smooth $u$71-admissible initial data, and structural monotonicity or barrier hypotheses. They prove long-time existence of a unique smooth $u$72-admissible solution and convergence to the smooth elliptic Neumann solution of

$u$73

with the same boundary data. If the monotonicity is strict, the convergence is exponentially fast (Chen et al., 2020). In the time-independent Neumann case, the solution may instead converge to a translating profile

$u$74

Gong–Liu–Tu consider a broader Neumann class involving the vector-valued operator $u$75: $u$76 Under a Dong-type growth condition on $u$77,

$u$78

they derive interior gradient estimates, global a priori estimates, and existence of a unique $u$79-convex Neumann solution by the continuity method (Gong et al., 10 Jan 2025).

These works show that the quotient structure is compatible with both viscosity and classical approaches. Comparison principles, explicit barriers, generalized symmetric ansätze, continuity method, and parabolic smoothing all remain effective, but each requires admissibility to be encoded in a manner tailored to the underlying geometry and boundary condition.

5. Geometric formulations on manifolds and on the sphere

On closed Kähler manifolds, Sun studies the quotient equation

$u$80

for $u$81, with $u$82 unique up to an additive constant (Chen, 2021). The necessary and sufficient conditions are a cohomological cone condition $u$83 and the integral balance

$u$84

Under these conditions there exists a unique smooth solution with $u$85. The proof uses the continuity method together with $u$86, $u$87, and $u$88 a priori estimates, Evans–Krylov theory, and the necessity of the cone and integral conditions.

On compact HKT manifolds, Li Chen proves a $u$89 estimate for

$u$90

with no additional assumption on the hypercomplex structure beyond the HKT setting (Chen, 2022). The key ingredient is direct use of the cone condition

$u$91

combined with a Cherrier-type integration by parts argument and Moser iteration. The conclusion is a uniform $u$92 bound for normalized solutions.

On closed Riemannian manifolds, Sroka studies the real Hessian quotient equation with background tensor $u$93,

$u$94

In dimension two, for $u$95, he proves an unobstructed second-order estimate and a solvability theorem: if there exists some admissible function $u$96, then for every strictly positive $u$97 there is a unique admissible solution of

$u$98

on $u$99 (Sroka, 6 Jan 2025). The argument introduces the test function

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$00

and exploits fine concavity of $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$01. The paper explicitly contrasts the real case with the complex $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$02-equation, where known obstructions occur.

Guan–Sroka analyze the positive operator

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$03

for symmetric tensors $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$04 on Riemannian manifolds (Guan et al., 19 Sep 2025). They prove a special concavity estimate stronger than the classical Gårding concavity of $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$05, and from it derive a Jacobi inequality for $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$06 when $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$07 solves $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$08. This provides a general structural tool for real quotient equations on manifolds.

On the sphere, the $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$09 dual Christoffel–Minkowski-type problem considers a support function $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$10 on $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$11, with

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$12

and a quotient operator built from $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$13, the $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$14-eigenvalues of $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$15. The equation is

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$16

Under admissibility and structural conditions on $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$17, the paper proves a full rank theorem, $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$18, $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$19, and $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$20 a priori estimates, and existence and uniqueness of strictly spherically convex solutions (Luo et al., 13 Apr 2026). A central technical input is the “inverse convexity” property of the operator.

Taken together, these manifold formulations show that the positive Hessian quotient equation is not merely an Euclidean Hessian problem. It is also a cohomological, quaternionic, Riemannian, and convex-geometric equation, with admissibility encoded either by Gårding cones, cone conditions, positivity of support-function matrices, or inverse-convexity structures.

6. Structural identities, symmetry, and current landscape

Several recent advances rest on structural identities that control third derivatives or convert quotient equations into more tractable forms. Guan–Sroka’s special concavity estimate for $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$21 is one example; Zhou’s twisted special Lagrangian reformulation is another (Guan et al., 19 Sep 2025, Zhou, 2023). In the $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$22 case, Lu–Sroka’s elementary shift

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$23

transforms the quotient equation into a constant-right-hand-side $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$24-equation (Lu et al., 16 Feb 2026). These are not mere formal devices: they directly feed regularity and Liouville theory.

Overdetermined problems furnish a separate rigidity theory. Gao–Jia–Zhang consider

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$25

with $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$26 on $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$27. They prove that the only admissible solution is, up to translation,

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$28

using a Rellich–Pohozaev-type identity and the $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$29-function

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$30

together with the maximum principle (Gao et al., 2022). The result extends the Serrin-type theory from $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$31-Hessian equations to Hessian quotients.

The present landscape is therefore neither uniformly positive nor uniformly negative. On one side, there are complete Bernstein criteria (Du, 2021), semi-convex Liouville theorems (Lu et al., 16 Feb 2026), arbitrary-dimensional interior $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$32 estimates for certain $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$33 equations (Mei et al., 25 Apr 2026), and solvability theorems on Kähler manifolds and spheres (Chen, 2021, Luo et al., 13 Apr 2026). On the other side, there are explicit failures of interior $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$34 estimates for $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$35 when $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$36 (Lu, 2024). This rules out any naive expectation of a universal regularity theory for all quotient operators.

Several open directions are stated explicitly in the supplied literature. Lu–Sroka observe that generalizations of the $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$37 Liouville theorem to other $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$38 remain largely open, except in special cases such as $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$39 or special Lagrangian structures (Lu et al., 16 Feb 2026). Sroka formulates two conjectures in the Riemannian setting: unobstructed solvability for all positive right-hand sides, up to a multiplicative constant, and a general $\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$40 estimate

$\lambda(D^{2}u)=(\lambda_{1},\dots,\lambda_{n})$41

for admissible solutions (Sroka, 6 Jan 2025).

A plausible implication is that the future of the subject will depend less on a single universal technique and more on identifying operator-specific structures—special concavity, inverse convexity, Legendre duality, divergence-free tensors, or geometric reformulations—that isolate the regimes where quotient ellipticity can be converted into effective a priori control.