[Paper Review] On Uniqueness of Conformally Compact Einstein Metrics with Homogeneous Conformal Infinity

Gang Li|arXiv (Cornell University)|Dec 30, 2016

Geometric Analysis and Curvature Flows13 references3 citations

TL;DR

This paper establishes the uniqueness of non-positively curved conformally compact Einstein metrics on the 4-ball with a Berger metric on $S^3$ as conformal infinity, under the condition $\frac{1}{4} < \frac{\lambda_1}{\lambda_2} < 4$. Using asymptotic analysis of Einstein equations along geodesics from the center of gravity, and leveraging symmetry and Killing vector field extensions, the authors prove that such metrics are unique up to isometry and coincide with Pedersen's construction when curvature is non-positive.

ABSTRACT

In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^3, [\hat{g}])$ is close to that of the round sphere then any conformally compact Einstein manifold filled in must be negatively curved and simply connected, therefore if $\hat{g}$ is a Berger metric on $S^3$ with $Y(S^3, [\hat{g}])$ close to that of the round metric, the conformally compact Einstein metric filled in is unique up to isometries.

Motivation & Objective

• To establish global uniqueness of conformally compact Einstein metrics on the 4-ball with prescribed homogeneous conformal infinity.
• To extend conformal Killing vector fields on the boundary to Killing vector fields on the interior manifold using asymptotic Einstein metric expansions.
• To show that under symmetry and curvature constraints, the only such metric is Pedersen’s, up to isometry.
• To prove that when the Yamabe constant of the conformal infinity is close to that of the round sphere, the filled-in manifold is uniquely determined.

Proposed method

• Utilizes the geodesic defining function $x = Ce^{-r}$ from the center of gravity to reduce the Einstein equations to a boundary value problem along radial geodesics.
• Applies asymptotic expansion techniques for the Einstein metric to show that conformal Killing vector fields on $\partial M$ extend to asymptotically Killing vector fields on $M$.
• Employs a contradiction argument based on sign analysis of differences of solutions to the reduced ODE system, using the structure of the Einstein equations in radial form.
• Uses the existence of a unique center of gravity $p_0$ to ensure all isometries fix $p_0$, enabling reduction to radial symmetry.
• Analyzes the boundary value problem $(\ref{equn_BergerEinstein01})-(\ref{equn_BergerBV01})$ via integration and sign comparison of solution differences to prove uniqueness.
• Relies on Theorem 3.2 from Wang (2020) to guarantee extension of conformal Killing fields to Killing fields under non-positive curvature.

Experimental results

Research questions

• RQ1 Is there at most one non-positively curved conformally compact Einstein metric on the 4-ball with a Berger metric as conformal infinity?
• RQ2 Can conformal Killing vector fields on the boundary be extended to Killing vector fields on the interior manifold under curvature and symmetry assumptions?
• RQ3 Does the uniqueness of the solution to the reduced ODE system imply global uniqueness of the Einstein metric?
• RQ4 Under what conditions on the ratio $\lambda_1/\lambda_2$ is the conformally compact Einstein metric unique up to isometry?

Key findings

• For a Berger metric on $S^3$ with $\frac{1}{4} < \frac{\lambda_1}{\lambda_2} < 4$, there exists at most one non-positively curved conformally compact Einstein metric on the 4-ball, unique up to isometry.
• When $\frac{\lambda_1}{\lambda_2}$ is close to 1, the metric is unique and coincides with the perturbation solution from Graham and Lee (1991).
• The unique metric is explicitly given by Pedersen’s construction when the sectional curvature is non-positive.
• The extension of conformal Killing fields on $\partial M$ to Killing fields on $M$ is guaranteed by the asymptotic Einstein metric expansion and curvature assumptions.
• The boundary value problem $(\ref{equn_BergerEinstein01})-(\ref{equn_BergerBV01})$ admits at most one solution, proven via sign analysis of solution differences on intervals where derivatives do not vanish.
• The geodesic spheres centered at the center of gravity are homogeneous spaces invariant under the isometry group generated by Killing fields, due to homogeneous conformal infinity.