A priori estimates, blow-up limits, and degeneration for Abreu's equation on polytopes beyond the two-dimensional constant-right-hand-side case

Donaldson's theorem that a K-stable compact toric surface admits a constant-scalar-curvature Kähler metric rests on a priori estimates for Abreu's equation — the fourth-order equation $\sum_{i,j}\partial_i\partial_j u^{ij}=-A$ for the symplectic potential $u$ on the moment polytope, with Guillemin boundary conditions — proved only in dimension two and for constant right-hand side $A$. Donaldson's GAFA 2009 paper explicitly left open the extension to non-constant $A$ and higher dimensions, the classification of $C^0$ blow-up limits of rescaled solutions, a rigorous description of collapse when the stability condition fails, and sharp boundary exponents. The geometric existence question has since been resolved by other routes: Chen–Li–Sheng proved the two-dimensional prescribed-scalar-curvature case by direct convex analysis, and Chen–Cheng a priori estimates combined with convex analysis (Li–Lian–Sheng) give an if-and-only-if existence theorem on toric manifolds of every dimension. What remains open is the direct real-variable analysis of the boundary value problem itself: an $L^\infty$ bound on the cofactor vector field $V^i=-\sum_j\partial_j u^{ij}$ and boundary regularity in dimension $n\ge 3$, Donaldson's 'basic local question' of classifying $C^0$ limits of rescaled solutions, the conjectural crease/generalized-Taub–NUT degeneration picture, and the optimal boundary asymptotics (the exponent $\mu=1$). The problem below is this surviving analytic core, with the resolved parts documented in the background.

Background

A compact toric manifold is a compact complex Kähler manifold $X$ of complex dimension $n$ admitting an effective holomorphic action of the complex torus $(\mathbb{C}^*)^n$ with a dense open orbit. The moment map of the induced compact torus $T^n$-action has as image a convex polytope $P\subset\mathbb{R}^n$ which, for smooth $X$, is a Delzant polytope: at each vertex the primitive edge directions form a $\mathbb{Z}^n$-basis. By work of Guillemin and Abreu, the $T^n$-invariant Kähler metrics on $X$ in a fixed cohomology class correspond to their symplectic potentials: smooth strictly convex functions $u$ on $P$ satisfying the Guillemin boundary conditions. Writing $\ell_1,\dots,\ell_N$ for the affine functions defining the facets of $P$ (positive on $P$, with primitive integral gradient), the condition is that $u-\sum_a\ell_a\log\ell_a$ extends smoothly to the closure $\bar P$; the singular term prescribes the boundary behaviour of the potential. In these action-angle coordinates the scalar curvature of the metric equals $-\frac14(\sum_{i,j}\partial_i\partial_j u^{ij})\circ\pi$, where $(u^{ij})$ is the inverse of the Hessian matrix $(u_{ij})$ and $\pi:X\to P$ is the moment map. Hence the metric has prescribed scalar curvature $A$ exactly when $u$ solves Abreu's equation $\sum_{i,j}\partial_i\partial_j u^{ij}=-A\quad\text{on }P,$ with the Guillemin boundary conditions; constant $A$ gives a constant-scalar-curvature (cscK) metric and affine-linear $A$ an extremal metric in the sense of Calabi (a metric minimizing the $L^2$-norm of the scalar curvature over the Kähler class). Donaldson reduced this geometric problem to convex analysis on $P$. The natural variational object is the functional $\mathcal{M}(u)=-\int_P\log\det(u_{ij})\,d\mu+L_{A,\sigma}(u)$, where $L_{A,\sigma}(f)=\int_{\partial P}f\,d\sigma-\int_P Af\,d\mu$ and $\sigma$ is the boundary measure determined by the Guillemin data; its critical points with the Guillemin boundary conditions are exactly the solutions of Abreu's equation. Solvability forces $L_{A,\sigma}$ to be nonnegative on convex functions, i.e. matching of masses and moments of $(\partial P,\sigma)$ and $(P,A\,d\mu)$; Donaldson conjectured in 2002 that a quantitative version of this condition (uniform positivity, equivalently the polytope form of uniform K-stability — K-stability being the algebro-geometric stability condition, defined via degenerations of the polarized variety, that is conjecturally equivalent to cscK existence in the Yau–Tian–Donaldson picture; the Futaki invariant, a character of the Lie algebra of holomorphic vector fields that obstructs cscK existence, is its infinitesimal version) is also sufficient. His series of papers established interior estimates for Abreu's equation, a continuity method under an 'M-condition' (a uniform bound on difference quotients of directional derivatives across the polytope), and the key a priori Euclidean bound $|V|\le C$ for the vector field $V^i=-\sum_j\partial_j u^{ij}$, proved however only in dimension two and for constant $A$ (Theorem 2 of the continuity-method paper). The GAFA 2009 paper completed the sufficiency in that case. Its proof proceeds by contradiction and blow-up: rescaled convex functions converge locally uniformly, and the central difficulty — in the author's words, 'our lack of understanding of this basic local question' — is to describe which convex functions can arise as such $C^0$ limits. Along the way the edge analysis yields (Corollary 4) a lower bound $\Delta(t_1,t_2)\ge C_\mu(t_2-t_1)^\mu$ for every $\mu>1$, where $\Delta(t_1,t_2)=\int_{t_1}^{t_2}u_{22}\,dt$ along an edge; the optimal exponent $\mu=1$, equivalent to an intrinsic lower bound on $u_{22}$ along the edge, resisted proof. Section 6 of the paper describes, 'albeit somewhat conjectural[ly]', the expected behaviour when positivity fails: a crease $\{\lambda=0\}$ — the zero set of the affine function $\lambda$ with $L(\lambda^+)=0$, $\lambda^+=\max(\lambda,0)$ — divides $P$, the solutions normalize differently on the two sides and blow up like $n_\alpha\lambda^+$, and near a crease through a vertex the blow-up limits should be explicit complete scalar-flat metrics generalizing the Taub–NUT metric (the standard complete Ricci-flat $4$-metric with two-torus symmetry), with quarter-plane symplectic potential $u=x_1\log y_1+x_2\log y_2+\frac12(a_2y_1^2+a_1y_2^2)$, Ricci-flat only for $a_1=a_2$. The landscape has since changed in three ways, which delimit what is genuinely open. First, Chen–Li–Sheng resolved the two-dimensional problem with general prescribed scalar curvature (edge-nonvanishing smooth $A$) under uniform stability, by direct convex analysis with new boundary techniques. Second, Chen–Cheng proved a priori estimates for cscK-type equations on general compact Kähler manifolds, and combined with convex analysis on polytopes this yielded the if-and-only-if theorem of Li–Lian–Sheng: on a compact toric manifold of any dimension, relative K-polystability of the polytope data is equivalent to the existence of a smooth invariant metric with prescribed scalar curvature. The geometric existence problem is therefore closed — but by complex-geometric methods that bypass the Abreu equation, leaving its real-variable estimate theory untouched. Third, the boundary regularity problem for the decoupled second-order member of the Abreu system (the singular Monge–Ampère equation $\det D^2u=\ell_1\cdots\ell_N/H$ with Guillemin data) was settled in the polygon by Rubin and G. Huang and extended to high dimension by Huang–Shen, while for the fourth-order equation recent literature states that 'the regularity of the solutions for a general function $S$ in any dimension remains widely open'. The model metrics of the degeneration picture have been constructed, classified, and their $L^2$-curvature properties verified (Abreu–Sena-Dias, Sena-Dias, Weber), but their conjectural appearance as blow-up limits of degenerating polytope solutions is unproven. The open problem below is this surviving analytic core.

Problem Statement

Let $P\subset\mathbb{R}^n$ be a bounded Delzant polytope with facet-defining affine functions $\ell_1,\dots,\ell_N$, let $\sigma$ be the corresponding Guillemin boundary measure, and let $A$ be a smooth function on $\bar P$; let $u$ denote a smooth strictly convex solution of Abreu's equation $\sum_{i,j}\partial_i\partial_j u^{ij}=-A$ on $P$ satisfying the Guillemin boundary conditions ($u-\sum_a\ell_a\log\ell_a\in C^\infty(\bar P)$), normalized at a base point. Beyond Donaldson's two-dimensional, constant-$A$ theorem, the open problem is the direct real-variable analysis of this boundary value problem. Its common objective is the a priori estimate, compactness and degeneration theory of the equation, pursued through four linked but independently resolvable deliverables — (ii) is the analytical engine required by (i), (iii) is (ii) applied to degenerating data, and (iv) is the quantitative sharpening of the boundary control in (i). Deliverable (i) is the core question; each deliverable may be settled either by a proof of the stated claim or by an explicit counterexample, a solution of (i) alone resolves the main a priori-estimate problem while leaving (ii)–(iv) open, and (ii), (iii) and (iv) each stand as separate questions that may be settled independently of the others:

(i) A priori estimates in dimension $n\ge 3$ (and for non-constant $A$): prove an analogue of Donaldson's bound — an a priori Euclidean $L^\infty$ bound $\sup_P|V|\le C$ for the vector field $V^i=-\sum_j\partial_j u^{ij}$, with $C$ depending only on $(P,\sigma,A)$ and the stability constant — together with up-to-boundary regularity near facets and vertices (an analogue of the M-condition).

(ii) Donaldson's 'basic local question': classify the possible $C^0$ limits — given a sequence of smooth convex functions on a ball obtained as rescalings of solutions, so that $u^{ij}_{ij}\to 0$ in the weak sense and a fixed $C^0$ bound holds after normalization, determine exactly which convex functions can arise as locally uniform limits.

(iii) The degeneration picture when positivity fails: for convergent data $(P^{(\alpha)},\sigma^{(\alpha)},A^{(\alpha)})$ whose limit violates the positivity condition, with an affine function $\lambda$ changing sign on $P$ with $L(\lambda^+)=0$ (unique up to scale, in the basic case), determine whether the solutions converge with different normalizations on the two sides of the crease $\{\lambda=0\}$, blowing up like $n_\alpha\lambda^+$ with $n_\alpha\to\infty$, and whether near a crease through a vertex the rescaled limits are the generalized Taub–NUT scalar-flat models — the quarter-plane potentials $u=x_1\log y_1+x_2\log y_2+\frac12(a_2y_1^2+a_1y_2^2)$ — identifying which model metrics arise and the corresponding curvature behaviour, or establish that the conjectured picture fails.

(iv) Sharp boundary asymptotics for the fourth-order equation: decide whether the exponent in Corollary 4 of the GAFA 2009 paper can be taken to be the optimal value $\mu=1$ — equivalently, whether $u_{22}$ admits an intrinsic positive lower bound along an edge — and establish the optimal blow-up rates of the second derivatives of the potential $u$ (that is, of the Hessian $u_{ij}$, whose leading boundary singularity is that of the model $\sum_a\ell_a\log\ell_a$) near facets and vertices, in the manner recently obtained for the decoupled second-order Guillemin problem.

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