Nonexistence of harmonic diffeomorphisms from the disk onto a complete parabolic plane (Mazet–Rodríguez–Rosenberg conjecture)

Open problem stated at the end of the introduction of Mazet, Rodríguez and Rosenberg, 'Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms' (Amer. J. Math. 141, 2019): there is no harmonic diffeomorphism from the unit disk onto the plane $\mathbb{R}^2$ endowed with a complete parabolic metric, i.e. a complete metric conformally equivalent to the Euclidean metric. Nonexistence is currently known only under curvature control on the target — non-negative Gaussian curvature (Heinz; Schoen) or an at most logarithmic bound on the integrated negative curvature (Mazet–Rodríguez–Rosenberg) — while all known constructions of harmonic diffeomorphisms from simply connected surfaces produce targets of hyperbolic conformal type (Collin–Rosenberg; Gálvez–Rosenberg). Whether some complete parabolic metric on the plane can be the target of a harmonic diffeomorphism with domain the disk remains open.

Background

A harmonic map is a critical point of the Dirichlet energy; for a smooth map $u$ between oriented surfaces it is characterized by the vanishing of its tension field $\tau(u)=\operatorname{tr}\nabla du$, a second-order elliptic system. When the domain is a surface, harmonicity is invariant under conformal changes of the domain metric, so a 'harmonic diffeomorphism from the unit disk $\mathbb{D}$' is well defined once the disk's conformal structure and the target metric are fixed; the target metric itself, however, enters the equations through its Christoffel symbols, and a conformal change of the target metric does not preserve harmonicity. A Riemannian surface is called parabolic if it carries no Green's function (equivalently, Brownian motion on it is recurrent); parabolicity is a conformal invariant, so a metric on the topological plane is parabolic exactly when it is conformally equivalent to the Euclidean metric. A metric is complete when geodesics run for all time. The Euclidean plane is the model complete parabolic surface, while the hyperbolic plane $\mathbb{H}^2$, conformally a disk, is the model surface of hyperbolic (non-parabolic) type.

Whether the conformal types of two surfaces are constrained by the existence of a harmonic diffeomorphism between them goes back to Schoen and Yau's study of univalent (one-to-one) harmonic maps between surfaces. Heinz (1952), in work on the minimal surface equation, proved the flat case: no harmonic diffeomorphism from the disk onto the Euclidean plane; Schoen (1993) generalized this to all complete targets of non-negative Gaussian curvature. In the opposite direction, the conjecture of Schoen and Yau that no harmonic diffeomorphism exists from the complex plane $\mathbb{C}$ onto $\mathbb{H}^2$ was disproved by Collin and Rosenberg (2010): using the Jenkins–Serrin theory of minimal graphs with infinite boundary data, they built an entire minimal graph over $\mathbb{H}^2$ in $\mathbb{H}^2\times\mathbb{R}$, conformally $\mathbb{C}$, whose vertical projection is a harmonic diffeomorphism from $\mathbb{C}$ onto $\mathbb{H}^2$. Gálvez and Rosenberg (2010) extended such existence to Hadamard surfaces (complete, simply connected, non-positive curvature) with curvature $K\le a<0$, and Tam and Wan (1994) constructed harmonic diffeomorphisms into Cartan–Hadamard surfaces with prescribed Hopf differentials — the Hopf differential being the holomorphic quadratic differential canonically attached to a harmonic map between surfaces. So harmonic diffeomorphisms onto hyperbolic-type targets are plentiful, and onto flat or nearly flat complete targets they are impossible.

Mazet, Rodríguez and Rosenberg (2019) pushed on both sides. They extended the Jenkins–Serrin construction by the theory of divergence lines (curves on the target along which the constructed minimal graphs have infinite boundary values and become vertical), producing entire minimal graphs over hyperbolic surfaces of finite topology and infinite area and hence harmonic diffeomorphisms from parabolic minimal graphs onto hyperbolic targets — again hyperbolic targets. And they proved a nonexistence theorem generalizing Schoen's: if $g_1$ and $g_2$ are complete metrics on an orientable surface $S$ with compact boundary, the negative part $K^-_{g_2}=\max\{0,-K_{g_2}\}$ of the target curvature satisfies $\int_{S_r(g_2)}K^-_{g_2}\,d\sigma_{g_2}\le C\ln(2+r)$ for some constant $C$ and all $r>0$, and a harmonic diffeomorphism $(S,g_1)\to(S,g_2)$ exists, then $(S,g_1)$ is parabolic; in particular the conformal disk, being of hyperbolic type, admits no harmonic diffeomorphism onto any such target. What this leaves untouched is precisely the geometry in question: complete metrics on $\mathbb{R}^2$ of parabolic conformal type whose negative curvature is spread more abundantly than a logarithmic budget allows. For such targets no nonexistence technique is available (Heinz's, Schoen's and the Mazet–Rodríguez–Rosenberg arguments all consume curvature control), and no existence construction is available either (all known constructions produce targets of hyperbolic conformal type). The authors closed their introduction by conjecturing that completeness together with parabolic conformal type of the target suffices, with no curvature hypothesis at all.

Problem Statement

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with its standard conformal structure (equivalently, with the hyperbolic metric; the choice within the conformal class does not affect harmonicity of maps out of $\mathbb{D}$). Does there exist a smooth complete Riemannian metric $g$ on $\mathbb{R}^2$ of parabolic conformal type — that is, a metric conformally equivalent to the Euclidean metric on the plane — and a diffeomorphism $u:\mathbb{D}\to\mathbb{R}^2$ that is harmonic as a map into $(\mathbb{R}^2,g)$, i.e. whose tension field $\tau_g(u)=\operatorname{tr}\nabla du$, computed with the Levi-Civita connection of $g$, vanishes identically? The conjecture of Mazet, Rodríguez and Rosenberg asserts a negative answer: for every complete parabolic metric $g$ on $\mathbb{R}^2$, every harmonic map from the disk into $(\mathbb{R}^2,g)$ fails to be a diffeomorphism onto $\mathbb{R}^2$. The task is to prove this nonexistence for the full class of complete parabolic target metrics — with no curvature sign, curvature growth, rotational symmetry, or quasiconformality (i.e. bounded-distortion) hypothesis on the map — or to refute it by exhibiting one such metric together with a harmonic diffeomorphism of the disk onto it.

References

1. Laurent Mazet, Magdalena Rodríguez, Harold Rosenberg, 'Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms', American Journal of Mathematics 141 (2019), no. 5, 1149–1177. DOI: 10.1353/ajm.2019.0029, https://doi.org/10.1353/ajm.2019.0029 (preprint: arXiv:1607.05061, https://arxiv.org/abs/1607.05061)
2. Erhard Heinz, 'Über die Lösungen der Minimalflächengleichung', Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Math.-Phys.-Chem. Abt. 1952 (1952), 51–56. Zbl: 0048.15401, https://zbmath.org/?q=an%3A0048.15401
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6. José A. Gálvez, Harold Rosenberg, 'Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces', American Journal of Mathematics 132 (2010), no. 5, 1249–1273. DOI: 10.1353/ajm.2010.0001, https://doi.org/10.1353/ajm.2010.0001
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