[Paper Review] Barycenter technique and the Riemann mapping problem of Escobar

Martin Mayer, Cheikh Birahim Ndiaye|arXiv (Cornell University)|May 22, 2015

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper resolves the remaining open cases of Escobar's Riemann mapping problem for compact Riemannian manifolds with boundary of dimension $ n \geq 3 $, proving that every such manifold admits a conformal scalar flat metric with constant mean curvature on the boundary. By refining the barycenter technique of Bahri–Coron using Chen’s bubbles and algebraic topological arguments, the authors establish the existence of solutions to the critical boundary value problem via variational methods, completing the classification of all cases for $ n \geq 3 $. The key contribution is the resolution of the final open cases when $ n \geq 6 $, the boundary is umbilic, and the Sobolev quotient is positive.

ABSTRACT

We solve the remaining cases of the Riemann mapping problem of Escobar. Indeed, performing a suitable scheme of the barycenter technique of Bahri-Coron via the Chen's bubbles, we solve the cases left open after the work of Chen. Thus, combining our work with the ones of Almaraz, Chen, Escobar and Marques we have that every compact Riemannian manifold with boundary of dimension greater or equal than 3 carries a conformal scalar flat metric with constant mean curvature.

Motivation & Objective

• To resolve the remaining open cases of Escobar's Riemann mapping problem for compact Riemannian manifolds with boundary in dimensions $ n \geq 3 $.
• To establish the existence of a conformal scalar flat metric with constant mean curvature on the boundary for all compact $ n $-dimensional manifolds with $ n \geq 3 $.
• To complete the classification of cases left unresolved by prior works of Almaraz, Chen, Escobar, Marques, and others.
• To apply the barycenter technique via Chen’s bubbles and algebraic topological methods to prove existence of solutions to the critical boundary value problem.

Proposed method

• Adapting the barycenter technique of Bahri–Coron using Chen’s bubbles to handle concentration phenomena in the variational setting.
• Employing algebraic topological arguments via homology and the Gromov–Witten-type diagram to detect nontrivial critical points of the Escobar functional.
• Defining a suitable deformation retract and constructing a map $ f_p(\lambda) $ between configuration spaces of points on the boundary to analyze topological degree.
• Using the critical point theory of the Escobar functional $ \mathcal{E}_g(u) $, which is equivalent to solving the boundary value problem $ L_g u = 0 $ in $ M $, $ B_g u = 2(n-1)u^{n/(n-2)} $ on $ \partial M $.
• Applying the positivity of the Sobolev quotient $ \mathcal{Q}(M,\partial M,g) > 0 $ as a key assumption to ensure coercivity and avoid concentration degeneracy.
• Establishing commutative diagrams in homology to detect nontrivial topological degree, thereby proving existence of a minimax solution.

Experimental results

Research questions

• RQ1 Does every compact Riemannian manifold with boundary of dimension $ n \geq 3 $ admit a conformal scalar flat metric with constant mean curvature on the boundary?
• RQ2 What are the remaining open cases of Escobar’s Riemann mapping problem after prior works by Escobar, Marques, Almaraz, and Chen?
• RQ3 Can the barycenter technique be extended to handle the critical boundary nonlinearity in the scalar curvature problem with constant mean curvature?
• RQ4 Under what topological and geometric conditions does the Escobar functional admit a positive minimax solution?
• RQ5 Is the positivity of the Sobolev quotient sufficient to guarantee existence of a solution in the case of umbilic boundary and $ n \geq 6 $?

Key findings

• The paper proves that every compact Riemannian manifold with boundary of dimension $ n \geq 3 $ carries a conformal scalar flat metric with constant mean curvature on the boundary, thus fully resolving Escobar’s problem.
• The remaining open cases—specifically $ n \geq 6 $, umbilic boundary, and $ \mathcal{Q}(M,\partial M,g) > 0 $—are resolved via the barycenter technique and algebraic topology.
• The existence of a solution to the boundary value problem $ L_g u = 0 $ in $ M $, $ B_g u = 2(n-1)u^{n/(n-2)} $ on $ \partial M $ is established under the stated conditions.
• The proof relies on constructing a continuous map between configuration spaces of points on the boundary and showing that its induced homomorphism on homology is nontrivial, implying existence of a critical point.
• The key topological input is the nonvanishing of the induced map on homology $ (f_p(\lambda))_* $, which follows from a commutative diagram argument involving the boundary map and cup product structure.
• The result completes the classification initiated by Escobar, Marques, Almaraz, and Chen, showing that the only obstruction to existence is the positivity of the Sobolev quotient in the umbilic case.