How can polynomial complex Hessian equations be solved without symmetric eigenvalue assumptions?

Why the symmetric eigenvalue assumption is dropped

Classical fully non-linear elliptic Hessian equations on a compact Riemannian manifold X take the form f(λ(∇²u(x) + A(x))) = h(x), where λ denotes the eigenvalues of a symmetric endomorphism, f is concave and elliptic, and f is invariant under permutation of its variables. That permutation invariance is exactly what makes the equation spectral: it can be written as a symmetric function of the eigenvalues of the Hessian.

The paper's motivation is a family of polynomial complex Hessian equations on a compact Kähler manifold, written as a sum over k of γ_{n−k} ∧ χ^k / k!, where the γ_k are closed real (n−k, n−k)-forms and χ is a closed real (1,1)-form of fixed cohomology class. This family includes the Monge–Ampère equation, the J-equation and the deformed Hermitian Yang–Mills equation. The spectral case, where γ_k = c_k ω^k / k! for a Kähler form ω and constants c_k, was solved by Fu, Li, D. Zhang and Z. Zhang (at least when c_0 ≠ 0). The only previously solved non-spectral example of this form was the Fang–Ma equation, whose coefficients satisfy γ_0 > 0 with the other γ_k weakly non-positive.

Removing spectrality costs the concavity terms that the spectral theory relies on. The paper's response is to introduce the notion of an almost spectral equation, a technical condition depending on a tuple M of positive constants that controls the derivatives of a function F whose zero locus is the hypersurface H and recovers additional concavity terms.

Defining the equation as a level set, not a function

The paper's stated main contribution to the general theory is a change of viewpoint. Because the notion of a C-sub-solution is pointwise and depends only on the level sets {f = h(x)}, the author does not define the equation as a function of the Hessian but directly as the level set to be reached.

That level set must be a smooth Γ-elliptic convex hypersurface H inside a real Euclidean space E, where Γ ⊂ E is a non-empty open cone with strictly convex closure. Three conditions define it: H is a non-empty connected closed smooth hypersurface; at every A ∈ H the unit normal vector N_H(A) lies in the dual cone Γ^∨; and the second fundamental form of H is non-negative on the tangent space T_A H. The second condition is equivalent to saying that a linear form vanishing on T_A H is positive on all nonzero elements of Γ, up to sign, so it does not depend on the choice of scalar product. The third condition is equivalent to φ(B) ⩾ o(‖B − A‖²) as B ∈ H tends to A, also scalar-product independent.

The paper reports that this viewpoint allows more flexibility in the elliptic equations one can solve when a sub-solution is available, even in the spectral case. The equation actually solved is ∇²u(x) ∈ H_x for all x ∈ X, where ∇²u is the Hessian endomorphism of u with respect to g, viewed as a section of the bundle End_S(TX, g) of g-symmetric endomorphisms of the tangent bundle.

Building functions whose zero locus is the hypersurface

For each P ∈ Γ the paper constructs a smooth function F_P : E → R whose zero locus is H, using the implicit function theorem. Given A ∈ E, t ∈ R and P ∈ V with A − tP ∈ H and ⟨N_H(A − tP), P⟩ > 0, there is a unique continuous germ F_{A,t,P} with F_{A,t,P}(A, P) = t such that B − F_{A,t,P}(B, Q) Q ∈ H for all (B, Q) in a neighbourhood. Its gradient with respect to B at the origin is N_H(A − tP) / ⟨N_H(A − tP), P⟩.

The paper shows that these functions are concave and Γ-elliptic, meaning ∇F(A) ∈ Γ^∨ for all A ∈ E. It also shows that H cuts E into a bottom-left part H⁻ and a convex upper-right part H⁺, and proves Guan's inequality in this setting, which the paper says is very helpful for the C² estimates.

For the geometric setting E = End_S(TX, g) with Γ = End^{++}_S(TX, g), the open subset of positive definite endomorphisms, the paper defines families of Γ-elliptic hypersurfaces and then almost spectral families. The globally defined functions F_P are used both to obtain the a priori estimates and to set up the continuity method.

A priori estimates bound the C² norm

The first main result is an a priori estimate for the elliptic equation ∇²u(x) ∈ H_x. Here H_0 is a family of End^{++}_S(TX, g)-convex hypersurfaces such that for every x, H_{0,x} is an affine hyperplane contained in H⁻_x. Under this setup there is a constant C = C(X, g, H_0, δ, R, M) such that any solution u with max_X u = 0 satisfies ‖u‖_{C²} ⩽ C.

The estimate is assembled from several pieces. The L¹ estimate is described as very standard and follows the methods of Tosatti and Weinkove. The C⁰ estimate is also standard and uses Błocki's version of the Alexandroff–Bakelman–Pucci maximum principle. The C¹ estimate is obtained directly, rather than as a consequence of the C² estimate plus a Liouville-type theorem, by introducing an auxiliary function and studying what happens at a point where its maximum is reached. The C² estimate also uses an auxiliary function and follows the method of many authors; the additional concavity terms coming from almost spectrality are used here.

Once the C² norm is controlled, the Evans–Krylov theory and elliptic regularity give bounds on derivatives of u of arbitrary order. The paper also proves C^{2,α} a priori estimates, using the estimate of Caffarelli and Cabré for elliptic equations that depend on the variable x. Uniform ellipticity is quantified by a positive non-increasing function ν(t) = min{λ_min(N_{H_x}(A)) : (x, A) ∈ End_S(TX, g), ‖A‖ ⩽ t}, and the resulting constants α and C depend on X, g, ν(‖∇²u‖_{C⁰}) and a constant M′₁₃.

Existence by continuity under a cohomological condition

The second main result is an existence theorem. Let H ⊂ End_S(TX, g) be an almost spectral family of smooth End^{++}_S(TX, g)-elliptic convex hypersurfaces, and assume there exists a C-sub-solution u : X → R. Assume moreover that for all smooth u and h : X → R, if ∇²u(x) − h(x) Id ∈ H_x for all x ∈ X, then the integral of h over X with respect to the volume form vanishes. Under these hypotheses there is a smooth u : X → R, unique up to a constant, such that ∇²u(x) ∈ H_x for all x.

The paper notes that this additional integral condition can be obtained by a cohomological argument for some complex Hessian equations, but that it need not hold in general. The proof sets up a continuity method with the family of equations F(x, ∇²u(x)) = (1 − t) F(x, 0) for t ∈ [0, 1], where F = F_Id. The set I of t for which a smooth solution exists contains 0 because u = 0 solves the t = 0 equation. Openness follows from the implicit function theorem applied to a map into the Hölder space of C^{0,1/2} functions with vanishing integral, whose linearization is an isomorphism by ellipticity. Closedness follows from the uniform C^{2,α} a priori estimate and compactness of the closed balls of C^{2,α} in C^{2,α/2}, with a standard bootstrapping argument giving smoothness.

Uniqueness is proved separately. If H_1 and H_2 are smooth End^{++}_S(TX, g)-elliptic convex hypersurfaces with H_1 ⊂ (H_2)⁻, and u_1, u_2 are smooth functions with ∇²u_i(x) ∈ H_{i,x} for each i, then u_1 − u_2 is constant. The argument uses a linear elliptic operator □ whose kernel consists of constants, together with the fact that □u ⩾ 0 and an integration-by-parts identity forcing equality.

Solving for an extra unknown parameter

The third main result handles the case where the hypersurface family carries an additional unknown. Let a < b be real numbers and let H be a family of smooth (End^{++}_S(TX, g) × [a, b])-elliptic convex hypersurfaces in the bundle End_S(TX, g) × [a, b] → X × [a, b]. For each c ∈ [a, b], the slice H^c = {(x, A, t) ∈ H : t = c} is a family of smooth End^{++}_S(TX, g)-elliptic convex hypersurfaces on End_S(TX, g). The paper assumes a uniform tuple M such that each H^c is almost spectral with that tuple, and that the map c ↦ (H^c_p)⁻ is non-decreasing for inclusion.

It also assumes a smooth u : X → R that is a C-sub-solution for H^b and satisfies ∇²u(x) ∈ (H^a_x)⁺ ∩ (H^b_x)⁻ = ⋃_{c ∈ [a,b]} H^c_x for all x ∈ X. Under these assumptions there is a smooth u : X → R, unique up to a constant, and a c ∈ [a, b] such that ∇²u(x) ∈ H^c_x for all x. If in addition H^{c_1} ∩ H^{c_2} = ∅ whenever c_1 ≠ c_2, then c is unique too.

The proof runs a two-parameter continuity method over (t, c) ∈ [0, 1] × [a, b] with equations F^c(x, ∇²u(x)) = (1 − t) F^c(x, 0) + K, where the unknown is the pair (u, K) with K a real constant. A lemma bounds K between min_X F^b(x, 0) and max_X F^a(x, 0). The argument then tracks the sets J_1 = {t ∈ [0, 1] : (t, b) ∈ I} and J_2 = {c ∈ [a, b] : (1, c) ∈ I}, showing J_1 = [0, 1] and splitting the analysis of J_2 into two cases depending on the sign of K_{1,c}.

What the theorems do and do not cover

The results are stated for a compact Riemannian manifold (X, g) of dimension n and for hypersurfaces in the bundle of g-symmetric endomorphisms of the tangent bundle. The existence theorems require an almost spectral family, which is a technical assumption depending on a tuple M of positive constants, and they require a C-sub-solution. Theorem 2 additionally requires the integral condition on h; the paper explicitly says this condition can be obtained by a cohomological argument for some complex Hessian equations but need not hold in general. Theorem 3 instead assumes a C-sub-solution for H^b together with the intersection condition ∇²u(x) ∈ (H^a_x)⁺ ∩ (H^b_x)⁻, and it produces both a solution and a parameter c, with uniqueness of c only under the extra disjointness assumption H^{c_1} ∩ H^{c_2} = ∅ for c_1 ≠ c_2.

The paper is the first in a series whose stated purpose is to give a numerical criterion to solve polynomial Hessian equations on a compact Kähler manifold, aiming at assumptions on the coefficients γ_k that cover both the Fu–Li–Zhang–Zhang spectral equations and the Fang–Ma non-spectral equation. The present article develops the analytic framework — the general existence theorem for non-spectral Hessian equations on compact Riemannian manifolds — rather than delivering that final numerical criterion. The paper also states that AI was used marginally to correct spelling, polish the introduction and generate LaTeX code for the figures, with the author taking responsibility for all mathematical content.

The practical distinction for a reader is therefore between the two existence routes. Theorem 2 keeps the hypersurface family fixed and pays for existence with the integral condition on h, which the paper says is not guaranteed in general. Theorem 3 avoids that condition by letting the parameter c be unknown, but it requires a C-sub-solution for the upper family H^b and the intersection condition on ∇²u, and it only pins down c when the slices H^c are pairwise disjoint. Which route applies depends on whether the specific polynomial complex Hessian equation at hand supplies the cohomological vanishing needed by Theorem 2 or instead fits the monotone family structure of Theorem 3.

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