[Paper Review] Prescribed Scaler Curvatures for Homogeneous Toric Bundles

Bohui Chen, Qing‐Long Han+3 more|arXiv (Cornell University)|Mar 7, 2016

Geometry and complex manifolds15 references3 citations

TL;DR

This paper establishes the existence of Kähler metrics with prescribed scalar curvature on homogeneous toric bundles over compact toric surfaces by solving a generalized Abreu equation on Delzant polytopes. Under uniform stability and edge-nonconstant Duistermaat-Heckman polynomial conditions, the authors prove $C^{6,\alpha}$ regularity and construct a smooth $G$-invariant metric with scalar curvature $\mathbb{S} = A + h_G$, affirming the Yau-Tian-Donaldson conjecture in this setting for $n=2$.

ABSTRACT

In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.

Motivation & Objective

• Address the existence of Kähler metrics with prescribed scalar curvature on homogeneous toric bundles, extending results from toric varieties.
• Generalize the Abreu equation to include geometric data $\mathbb{D}$ (Duistermaat-Heckman polynomial) and $h_G$ (group contribution) for toric fibrations.
• Establish boundary regularity for the generalized Abreu equation in dimension two, particularly near edges and vertices of the Delzant polytope.
• Prove that uniform stability and edge-nonconstant $\mathbb{D}$ are sufficient conditions for the existence of smooth solutions with prescribed scalar curvature.
• Provide a constructive solution to the Yau-Tian-Donaldson conjecture for homogeneous toric bundles in complex dimension two.

Proposed method

• The generalized Abreu equation is formulated as $-\frac{1}{\mathbb{D}}\sum_{i,j=1}^{2}\frac{\partial^2(\mathbb{D}u^{ij})}{\partial\xi_i\partial\xi_j} = \mathbb{S} - h_G$ on a Delzant polytope $\Delta \subset \mathbb{R}^2$, where $\mathbb{S}$ is the scalar curvature and $\mathbb{D}, h_G$ are known geometric functions.
• Boundary regularity is established using affine blow-up techniques, differential inequalities on affine-invariant quantities, and the Bernstein theorem to control behavior near edges.
• Near vertices, subharmonic functions are constructed in preimages of neighborhoods under the moment map to control regularity at corners.
• Uniform $C^{6,\alpha}$ estimates are derived via $G$-invariance and bounds on $\|\nabla \log \mathbb{F}_{f_\vartheta}\|_{\mathcal{G}_f}$, ensuring convergence of approximate solutions.
• Key estimates involve the ratio of determinants of Hessian matrices and lower bounds on geodesic distances in the metric structure.
• Stability conditions are applied via the uniform stability definition (Definition 2.1), ensuring solvability and regularity of the solution $\phi$ to the equation.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions does the generalized Abreu equation admit a smooth solution on a Delzant polytope for homogeneous toric bundles?
• RQ2 Can the method of Chen, Li, and Sheng for the Abreu equation be extended to handle the generalized case with nontrivial $\mathbb{D}$ and $h_G$?
• RQ3 Does uniform stability and edge-nonconstant $\mathbb{D}$ imply $C^{6,\alpha}$ regularity of the solution to the generalized Abreu equation?
• RQ4 How does the presence of $h_G$ affect the scalar curvature structure and the solvability of the equation compared to the classical Abreu case?
• RQ5 Is the Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics valid in the setting of homogeneous toric bundles over toric surfaces?

Key findings

• The generalized Abreu equation admits a unique smooth solution $\phi \in C^{6,\alpha}(\bar{\Delta})$ under the assumptions of uniform stability and edge-nonconstant $\mathbb{D}$.
• A smooth $G$-invariant metric $\mathcal{G}$ exists on $G \times_K M$ such that its scalar curvature is $\mathbb{S} = A + h_G$, where $A$ is a given smooth function.
• Boundary regularity near edges and vertices is achieved through a combination of Ricci curvature estimates, geodesic distance bounds, and subharmonic function constructions.
• Uniform $C^{6,\alpha}$ bounds are established on the potential function $f$, ensuring convergence of approximate solutions and smoothness of the resulting metric.
• The eigenvalues of the metric tensor and its Hessian are uniformly bounded above and below, ensuring non-degeneracy and regularity of the solution.
• The solution satisfies $\|f_\vartheta\|_{C^{6,\alpha}(U)} \leq C_5$ for any compact subset $U \subset \subset \Omega_\vartheta$, with $C_5$ depending on $\|A\|_{C^3(\Delta)}$ and geometric data.