[Paper Review] Potential Theory on Gromov Hyperbolic Manifolds of Bounded Geometry

Matthias Kemper, Joachim Lohkamp|arXiv (Cornell University)|May 6, 2018

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper reinterprets and streamlines Alano Ancona's potential theory on Gromov hyperbolic manifolds of bounded geometry, making it accessible to a broader audience by relying on coarse geometric and analytic invariants. It establishes that on such spaces, the Martin boundary coincides with the Gromov boundary, and proves sharp boundary Harnack inequalities that characterize uniform domains in $\mathbb{R}^n$, linking geometric hyperbolicity to analytic control of harmonic functions.

ABSTRACT

In the 1980s Alano Ancona developed a profound potential theory on Gromov hyperbolic manifolds of bounded geometry. Since then, such hyperbolic spaces have become basic in geometry, topology and group theory. In this paper we make Ancona's original work, addressed to a rather advanced audience, approachable for a wider readership without firm background in potential theory. We streamline the arguments to make the results depend only on coarse geometro-analytic invariants and we derive useful details not found in the original sources. We explain remarkable relations of these results to recent developments in the potential theory and quasi-conformal geometry of Euclidean domains.

Motivation & Objective

• To make Ancona's advanced potential theory on Gromov hyperbolic manifolds accessible to researchers without deep background in potential theory.
• To reframe results using only coarse geometric and analytic invariants—such as hyperbolicity and coercivity constants—rather than advanced analytic tools.
• To establish a sharp connection between geometric hyperbolicity and analytic control, particularly through boundary Harnack inequalities.
• To demonstrate how conformal deformation into hyperbolic spaces transfers potential-theoretic estimates to Euclidean domains.
• To show that uniform domains in $\mathbb{R}^n$ are characterized precisely by the validity of boundary Harnack inequalities for the Laplacian.

Proposed method

• Uses Gromov hyperbolicity and bounded geometry to derive local Harnack inequalities and Green's function estimates via resolvent equations.
• Applies $\varPhi$-chains to control Green's function growth and prove the main technical result: Theorem 5.1 on Green's functions along chains.
• Employs balayage and Martin theory to relate solutions of elliptic operators to boundary behavior, especially at the Gromov boundary.
• Applies Gromov uniformization to transform Euclidean uniform domains into hyperbolic manifolds via conformal deformation, preserving solution structure.
• Uses the Stein-Whitney smoothing procedure to construct a $\alpha_n \cdot \delta_c$-hyperbolic metric of bounded geometry from a uniform domain.
• Transfers boundary Harnack inequalities from the hyperbolic setting back to the original Euclidean domain using the conformal invariance of the Laplacian and Green's functions.

Experimental results

Research questions

• RQ1 How can Ancona's potential theory on Gromov hyperbolic manifolds be re-expressed using only coarse geometric and analytic invariants?
• RQ2 What is the precise relationship between the Martin boundary and the Gromov boundary on Gromov hyperbolic manifolds of bounded geometry?
• RQ3 To what extent does Gromov hyperbolicity ensure the validity of boundary Harnack inequalities for elliptic operators?
• RQ4 How do conformal deformations of Euclidean domains into hyperbolic spaces preserve potential-theoretic estimates?
• RQ5 Can the boundary Harnack inequality for the Laplacian on a domain in $\mathbb{R}^n$ characterize the uniform domain condition?

Key findings

• On Gromov hyperbolic manifolds of bounded geometry, the Martin boundary coincides with the Gromov boundary, and all Martin boundary points are minimal.
• The boundary Harnack inequality holds with a constant $C(a, \varepsilon, c, n) \geq 1$ depending only on geometric parameters, not on the specific domain.
• For uniform domains in $\mathbb{R}^n$, the Martin boundary for the Laplacian is homeomorphic to the topological boundary, and the boundary Harnack inequality characterizes uniformity.
• The Green's function on a uniform domain satisfies $G(x,y) = \delta_c^{n-2}(y) G^{\mathrm{Eucl}}(x,y)$ under conformal deformation, preserving solution structure.
• Non-uniform domains—such as certain cones or rotationally symmetric domains—can have uncountably many Martin boundary points or a Martin boundary homeomorphic to $S^{n-2}$, showing breakdown of Harnack control.
• The Herglotz integral representation for positive harmonic functions on the disk is recovered as a Martin integral over the Gromov boundary, with $K_\zeta(x) = \frac{1-|x|^2}{|x-\zeta|^2}$.