[Paper Review] Conformal Geometry on Four Manifolds

Sun‐Yung A. Chang|arXiv (Cornell University)|Sep 17, 2018

Geometric Analysis and Curvature Flows53 references3 citations

TL;DR

This paper presents a comprehensive survey of conformal geometry on four-dimensional manifolds, focusing on the role of the $σ_2$ curvature invariant and its connection to the Gauss-Bonnet-Chern formula, Paneitz operator, and $Q$-curvature. It establishes compactness results for conformally compact Einstein 4-manifolds under integral and curvature-based conditions, proving that boundedness of the $C^1$ norm of curvature and controlled boundary behavior imply $C^{k+2,\alpha}$ compactness up to diffeomorphism, with key applications to characterizing $S^4$ and $\mathbb{CP}^2$ via $\int \sigma_2$.

ABSTRACT

In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral conformal invariants on 4-manifolds and geometric applications. The development was heavily influenced by many earlier pioneer works; recent progress in conformal geometry has also been made in many different directions, here we will only present some slices of the development.

Motivation & Objective

• To survey the development of conformal geometry on 4-manifolds, particularly focusing on integral conformal invariants and their geometric applications.
• To characterize the diffeomorphism type of $(S^4, g_c)$ and $(\mathbb{CP}^2, g_{FS})$ using the integral of $\sigma_2$ over the manifold.
• To establish compactness results for sequences of conformally compact Einstein 4-manifolds under curvature and boundary curvature constraints.
• To relate the renormalized volume of CCE manifolds to the integral of $\sigma_2$ and explore conditions for existence and compactness.

Proposed method

• Utilizes the Gauss-Bonnet-Chern formula to express the Euler characteristic in terms of $\sigma_2$, the Weyl tensor, and scalar/Ricci curvatures on 4-manifolds.
• Applies variational methods to the $Q$-curvature and Paneitz operator $P_4$, linking them to $\sigma_2$ and the $4$-th order conformal invariants.
• Introduces a third-order boundary operator $P_3$ and curvature $T$ on the boundary of compact 4-manifolds with boundary, extending the conformal structure.
• Analyzes conformally compact Einstein (CCE) 4-manifolds by relating their renormalized volume to $\int \sigma_2$ and studying blow-up limits via rescaling.
• Employs Gromov-Hausdorff compactness and blow-up analysis to prove uniform $C^1$ bounds on curvature, leading to $C^{k+2,\alpha}$ compactness under boundary conditions.
• Applies Liouville-type PDE arguments to the limiting metric $g_\infty$, showing it must be the Poincaré metric $g_\mathbb{H}$, leading to contradiction if curvature norms are unbounded.

Experimental results

Research questions

• RQ1 Under what conditions is the family of metrics $\{g_i^*\}$ on a 4-manifold with boundary compact in $C^{k+2,\alpha}$ norm up to diffeomorphism?
• RQ2 How does the positivity of $\int_X \sigma_2(A_g)\,dv_g$ relate to the renormalized volume and compactness of CCE manifolds?
• RQ3 What role does the boundary curvature $T$ play in ensuring compactness of sequences of CCE metrics?
• RQ4 Can the perturbation condition in Theorem 5.1 be weakened from $\int_{B^4} \sigma_2 \geq 0$ to $\int_{B^4} \sigma_2 > 0$?
• RQ5 Is there a direct connection between the $L^1$-concentration of the $S$-tensor and the compactness of $\{g_i^*\}$ under the $T$-curvature condition?

Key findings

• The $C^1$ norm of the curvature of $\{g_i^*\ olimits\}$ is uniformly bounded under the assumption that $\int_{B^4} \sigma_2 \geq 0$ and the boundary $T$-curvature satisfies $\liminf_{r\to 0} \inf_i \inf_{S^3} \oint_{\partial B(x,r)} T_i \geq 0$.
• The family $\{g_i^*\}$ is compact in $C^{k+2,\alpha}$ norm for any $\alpha \in (0,1)$ and $k \geq 5$ if the $T$-curvature condition (5.6) holds.
• Compactness in $C^{k+2,\alpha}$ for $k \geq 2$ is achieved if the $S$-tensor satisfies $\lim_{r\to 0} \sup_i \sup_x \oint_{\partial B(x,r)} |S_i| = 0$, indicating no $L^1$ concentration.
• The limiting metric $g_\infty$ in the blow-up analysis is isometric to the Poincaré metric $g_\mathbb{H}$, and $\bar{w}_\infty = \log y$, leading to a contradiction if curvature norms are unbounded.
• The renormalized volume of a CCE manifold is equal to the integral of $\sigma_2$ over the manifold, linking geometric invariants to analytic quantities.
• The compactness results are essential steps toward an existence theory for CCE manifolds, particularly under controlled boundary and curvature conditions.