Guillemin Boundary Conditions

• Guillemin boundary conditions are prescribed logarithmic singularities along polytope facets that define canonical symplectic potentials in toric Kähler geometry.
• They underpin the analysis of singular fully nonlinear Monge–Ampère equations using techniques like barrier functions, rescaling, and weighted Hölder estimates.
• Their formulation bridges analytic PDE methods and geometric constructions, leading to robust existence, uniqueness, and regularity results in convex polyhedral settings.

Guillemin boundary conditions arise in the analysis of singular fully nonlinear partial differential equations on convex polytopes, most notably in the context of the Monge–Ampère equation and toric Kähler geometry. The hallmark of these boundary conditions is a prescribed logarithmic singularity of the solution along each codimension-one face (facet) of the polytope, matching the canonical behavior required for the symplectic potential of smooth toric Kähler metrics. Both the analytic structure and the geometric underpinning of Guillemin boundary conditions have been extensively developed, yielding robust existence, uniqueness, and regularity theorems for singular Monge–Ampère equations in polyhedral domains of arbitrary dimension (Rubin, 2014, Huang et al., 2024, Bayrami-Aminlouee et al., 27 Jun 2025).

1. Origin and Canonical Formulation

Guillemin boundary conditions were first identified as the asymptotic behavior needed for the symplectic potential on the moment polytope of a smooth compact toric Kähler manifold. In this construction, the polytope $P \subset \mathbb{R}^{n}$ has the representation

$$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$

where each $\ell_i$ is an affine defining function for the facet $F_i$. Guillemin proved that the symplectic potential $u_P$ necessarily takes the "model" form

$$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$

which exhibits a universal $\ell_i(x)\log\ell_i(x)$ singularity near each facet. In general, a strictly convex function $u$ satisfies the Guillemin boundary condition on $P$ if

$$u(x) - \sum_{i=1}^N \ell_i(x)\log\ell_i(x) \in C^{\infty}(\overline{P}). \tag{G}$$

This condition rigidly prescribes the leading order non-smoothness along $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$0 and is exactly what is required for the extension of Kähler metrics in the toric setting (Huang et al., 2024).

2. Monge–Ampère Equations with Guillemin Boundary Data

The principal PDE associated to Guillemin boundary data is the singular Monge–Ampère equation:

$$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$1

with $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$2, $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$3. The corresponding Guillemin boundary-condition is imposed via requirement (G), i.e., $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$4 (Huang et al., 2024). In the polygonal ($$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$5) case, the right-side may also be allowed to be merely Hölder continuous and strictly positive, which still yields solvability under suitable compatibility conditions at the vertices (Bayrami-Aminlouee et al., 27 Jun 2025).

The necessity for the singular denominator $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$6 is rooted in the asymptotics of the Monge–Ampère operator near the facets. For $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$7 approaching the $$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$8-th facet ($$P = \{ x \in \mathbb{R}^{n}: \ell_i(x) > 0, \ i=1, \ldots, N \},$$9), the blow-up of $\ell_i$0 as $\ell_i$1 is exactly matched by the model Guillemin expansion. This exact cancellation justifies both the PDE formulation and the boundary data (Rubin, 2014).

The solvability of such equations requires vertex compatibility conditions for $\ell_i$2, explicitly at each vertex $\ell_i$3 where $\ell_i$4. The compatibility is:

$\ell_i$5

where $\ell_i$6 are the facet normals (Huang et al., 2024).

3. Regularity and Asymptotics Near the Boundary

The analytic program centers on boundary regularity of $\ell_i$7 minus the Guillemin model, $\ell_i$8. The solution’s regularity is stratified by the local geometry:

• Near a facet (codim 1): After flattening, the model problem reduces to

$\ell_i$9

with $F_i$0 smooth, and $F_i$1 admits an expansion

$F_i$2

with $F_i$3 due to the matching with the Monge–Ampère singularity (Rubin, 2014).

• Near an edge or vertex (higher codimension): After subtracting $F_i$4 for $F_i$5 vanishing coordinates, the Taylor remainder is shown by inductive barrier and rescaling arguments to be smooth up to the boundary (Huang et al., 2024).

Weighted Hölder and Schauder estimates in terms of the local degenerate geometry (e.g., powers of $F_i$6 for $F_i$7-codimension faces) control the $F_i$8 and further regularity up to $F_i$9. In dimensions $u_P$0, this systematic approach (including the use of the Legendre transform, barrier functions, and the so-called D/E constants due to Donaldson) facilitates precise estimates for the Hessian and higher derivatives (Bayrami-Aminlouee et al., 27 Jun 2025).

4. Existence, Uniqueness, and Regularity in Arbitrary Dimension

General existence and uniqueness results, as established by Huang and Shen, show that if $u_P$1 is strictly positive and satisfies the vertex compatibility conditions, for any prescribed set of vertex values $u_P$2 there exists a unique convex $u_P$3 with

$u_P$4

The proof strategy combines the Perron–Aleksandrov method for construction of solutions, a priori barriers to guarantee convexity and boundary continuity, inductive exploitation of dimension-reducing flattenings, and regularity-boosting bootstraps up to every stratum of $u_P$5 (Huang et al., 2024).

In two dimensions and with lower regularity in $u_P$6, there are weighted regularity results ("Schauder-type") showing that solutions $u_P$7 are in a weighted Hölder space $u_P$8, where $u_P$9 is the singular metric induced by the model potential (Bayrami-Aminlouee et al., 27 Jun 2025). Near vertices, the Hessian of $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$0 has a controlled singularity matching $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$1 for local coordinates $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$2 vanishing at the corner.

5. Methodologies and Barrier Techniques

The fundamental analytic techniques for handling Guillemin boundary conditions are:

• Barrier functions: In neighborhoods of singular strata, model barriers of the form

$$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$3

trap the solution and force smoothness of the remainder (Huang et al., 2024).

• Rescaling and blow-up analysis: Local anisotropic rescalings around faces or vertices, together with Donaldson’s $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$4- and $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$5-constants, ensure that all possible blow-up limits must be degenerate and are excluded by convexity and boundary asymptotics (Bayrami-Aminlouee et al., 27 Jun 2025).
• Partial Legendre transforms: Near a facet, transforming tangential coordinates allows reduction to PDEs with coefficients holomorphic in the normal direction, giving access to classical Schauder interior estimates after degenerate rescaling (Rubin, 2014).
• Weighted Hölder/Schauder spaces: Regularity is measured in norms adapted to degeneracy at the boundary, e.g., $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$6 norms scaling by products of vanishing coordinates. These are essential to propagate $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$7 regularity of the remainder up to all faces (Huang et al., 2024, Bayrami-Aminlouee et al., 27 Jun 2025).

6. Applications in Toric Kähler and Affine Geometry

Guillemin boundary conditions are indispensable in toric Kähler geometry. The moment polytope $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$8 encodes the complex geometry of the toric manifold $$u_P(x) = \sum_{i=1}^N \ell_i(x) \log \ell_i(x) + f(x), \qquad f \in C^{\infty}(\overline{P}),$$9, and the symplectic potential $\ell_i(x)\log\ell_i(x)$0 with Guillemin boundary singularities yields a smooth Kähler metric on $\ell_i(x)\log\ell_i(x)$1. Analytically, the Monge–Ampère equation with right side $\ell_i(x)\log\ell_i(x)$2 and Guillemin boundary data constructs canonical metrics including toric cscK and Kähler–Einstein metrics by solving Abreu-type equations for prescribed curvature (Huang et al., 2024).

The boundary regularity theory for Guillemin-type PDEs additionally applies to degenerate elliptic equations in polyhedral domains, including eigenvalue-type Monge–Ampère equations and equations for affine spheres (Huang et al., 2024). The analytic machinery of barrier functions, singular expansions, and weighted regularity is transferrable to these settings.

7. Comparison with Other Symplectic Boundary Conditions

Earlier "symplectic" boundary conditions for differential forms (e.g., Tseng–Wang) are distinct from the Guillemin prescription. The Tseng–Wang conditions are formulated for primitive forms on symplectic manifolds with boundary, involving first- and second-order natural boundary operators in the symplectic Laplacian complex, and are strictly weaker than full Dirichlet/Neumann while still yielding an elliptic Hodge theory adapted to symplectic geometry (Tseng et al., 2017). In contrast, Guillemin boundary conditions are scalar, nonlinear, and dictated by the geometric requirement of smooth metric extension, specifically for the Monge–Ampère equation.

References Table

Key Paper	Main Content	arXiv ID
Monge–Ampère Equation with Guillemin Boundary Conditions	Existence/regularity in polytopes, asymptotic expansions	(Rubin, 2014)
Monge–Ampère Equation with Guillemin Condition in High Dimension	Full existence/uniqueness, detailed barrier analysis	(Huang et al., 2024)
Boundary Estimates for Monge–Ampère in Polygons	Schauder-type estimates, low-regularity existence	(Bayrami-Aminlouee et al., 27 Jun 2025)
Symplectic Boundary Conditions and Cohomology	New symplectic boundary conditions, Hodge theory	(Tseng et al., 2017)

These results collectively establish that Guillemin boundary conditions both dictate and enable the optimal analytic theory for singular Monge–Ampère equations on convex polytopes, with consequences for the construction of canonical metrics and for broader classes of singular PDEs (Rubin, 2014, Huang et al., 2024, Bayrami-Aminlouee et al., 27 Jun 2025).