Harmonic diffeomorphisms between Euclidean and hyperbolic model spaces: the open non-symmetric and higher-dimensional cases of Schoen's problem

The source paper (Du and Fan, arXiv:1409.2118, 2014) proves existence and uniqueness, up to a dilation and rotation of Euclidean space, of rotationally symmetric orientation-reversing harmonic diffeomorphisms from punctured hyperbolic $n$-space onto punctured Euclidean $n$-space, and cites Schoen's question on harmonic diffeomorphisms between model spaces as motivation. The headline instance of that question — whether a harmonic diffeomorphism from the complex plane onto the hyperbolic disk exists — was resolved in 2010 (Collin and Rosenberg constructed one; the reverse direction is excluded by Heinz–Schoen-type rigidity). The surviving open core stated here is the model-space existence and uniqueness classification beyond rotational symmetry: whether harmonic diffeomorphisms between full Euclidean and hyperbolic $n$-space exist for any $n\ge 3$ in either direction (neither a construction nor a general nonexistence theorem is known), whether a not-necessarily-rotationally-symmetric harmonic diffeomorphism $R^n_*\to H^n_*$ exists (rotationally symmetric ones do not), and whether every harmonic diffeomorphism $H^n_*\to R^n_*$ is rotationally symmetric up to a similarity of $R^n$ (only rotationally symmetric uniqueness is known).

Background

A harmonic map $u:(M,g)\to(N,h)$ between Riemannian manifolds is a critical point of the Dirichlet energy $E(u)=(1/2)\int_M |du|^2$, equivalently a solution of the elliptic system $\tau(u)=0$, where the tension field $\tau(u)$ is the trace of the second derivative of $u$. A harmonic diffeomorphism is a harmonic map that is a smooth bijection with smooth inverse. When the domain is a surface, the energy is invariant under conformal rescaling of the domain metric, so for maps between surfaces harmonicity depends only on the conformal structure of the source and the metric of the target; this fails in dimension $n\ge 3$, where the full domain metric enters the equations. The classical model spaces are Euclidean $n$-space $R^n$ and hyperbolic $n$-space $H^n$ (the simply connected complete Riemannian $n$-manifold of constant sectional curvature $-1$); write $M_*$ for a model space with one point removed. Heinz (1952) proved there is no harmonic diffeomorphism from the unit disk onto the flat plane, a rigidity input to Bernstein's theorem for entire minimal graphs (the classical statement that every entire minimal graph in $R^3$ is a plane). Schoen and Yau (1978) initiated the systematic study of one-to-one harmonic maps between surfaces, and in lecture notes published in 1993 Schoen asked whether there exists a harmonic diffeomorphism from the complex plane $C$ onto the hyperbolic unit disk $D$, conjecturing that none does. Parabolicity made the conjecture plausible: a surface is parabolic if every bounded subharmonic function on it is constant; $C$ is parabolic while $D$ is not, and quasiconformal maps (homeomorphisms of bounded infinitesimal distortion) preserve parabolicity, so any harmonic diffeomorphism $C\to D$ would necessarily have unbounded distortion. This two-dimensional question is now completely answered in both directions. Collin and Rosenberg (2010) constructed harmonic diffeomorphisms from $C$ onto $H^2$, disproving the Schoen–Yau conjecture: for a minimal graph over a surface $\Sigma$ in the Riemannian product $\Sigma\times R$ the vertical projection onto $\Sigma$ is harmonic, and they built entire minimal graphs over $H^2$ in $H^2\times R$ — Scherk-type graphs over ideal geodesic polygons (geodesic polygons with vertices at infinity) taking alternating asymptotic values $\pm\infty$, in the framework of Jenkins–Serrin theory (the theory of minimal-graph Dirichlet problems with prescribed infinite boundary data) — whose conformal type is the plane $C$. Gálvez and Rosenberg (2010) extended the construction to every Hadamard surface (a complete simply connected surface of nonpositive curvature) whose Gaussian curvature is bounded above by a negative constant. Conversely, no harmonic diffeomorphism exists from $D$ onto flat $C$: by conformal invariance Heinz's theorem covers any conformal metric on the disk domain, Schoen (1993) generalized the nonexistence to complete targets of nonnegative curvature, and Mazet, Rodríguez and Rosenberg (2019) proved the sharp parabolicity theorem that a harmonic diffeomorphism onto a complete surface whose negative curvature part satisfies $\int K^- \le C\log(2+r)$ forces a parabolic domain. What remains unsettled is the same existence problem in the remaining generality. Rotational symmetry reduces the harmonic-map system between $R^n$ and $H^n$ to one singular ordinary differential equation for the radial profile; along this line Tachikawa (1988, nonexistence under a differential inequality on the angular part), Akutagawa and Tachikawa (1993, nonexistence under rotational nondegeneracy at infinity or polynomial growth of the dilatation, the pointwise distortion quotient measuring deviation from conformality), Ratto and Rigoli (1993, asymptotics of radial harmonic maps), Cheung and Law (2004, an initial-value approach), and Chen–Du–Fan (2013) and Du–Fan (2014, 2016) settled all rotationally symmetric cases: there is no rotationally symmetric harmonic diffeomorphism between the full spaces $R^n$ and $H^n$ in either direction; between the punctured spaces, an orientation-reversing (Jacobian determinant everywhere negative) rotationally symmetric harmonic diffeomorphism $H^n_*\to R^n_*$ exists and is unique up to a dilation and rotation of $R^n$ (Du–Fan), while no rotationally symmetric harmonic diffeomorphism $R^n_*\to H^n_*$ exists (Du–Fan). Nothing is known for $n\ge 3$ without symmetry. For orientation, Schoen's second 1993 problem — extending a quasisymmetric boundary homeomorphism (a circle or sphere homeomorphism admitting a quasiconformal extension) to a harmonic quasiconformal self-map of $H^n$ — has also been resolved: Li–Tam (1993) proved uniqueness and existence for diffeomorphic boundary data, Marković (2015) confirmed the conjecture for $n=3$, Marković (2017) for $n=2$, and Lemm–Marković (2018) for all $n\ge 3$ via the heat flow. The boundary-extension side of the model-space program is thus closed, and the Euclidean–hyperbolic pairs without symmetry hypotheses are the surviving open core.

Problem Statement

Let $R^n$ and $H^n$ denote Euclidean and hyperbolic $n$-space with their standard complete metrics ($n\ge 2$), and let $R^n_*$ and $H^n_*$ be these spaces with one point removed. Call a map between two such spaces rotationally symmetric if it is equivariant under the rotations of both spaces about the distinguished point (the puncture, or a fixed origin for the full spaces); two maps are equivalent up to a similarity of $R^n$ if they differ by composition with a similarity, i.e., a composition of a dilation, an orthogonal transformation, and a translation of $R^n$; a similarity relating two maps into $R^n_*$ must preserve $R^n_*$ and hence fix the puncture, in which case it reduces to a composition of a dilation and an orthogonal transformation about the puncture. The following cases are settled: the two-dimensional full-space pair (a harmonic diffeomorphism $C\to H^2$ exists by Collin–Rosenberg and Gálvez–Rosenberg, and none exists $H^2\to C$ by Heinz–Schoen–Mazet–Rodríguez–Rosenberg); and all rotationally symmetric cases (no rotationally symmetric harmonic diffeomorphism between full $R^n$ and $H^n$ in either direction; rotationally symmetric orientation-reversing harmonic diffeomorphisms $H^n_*\to R^n_*$ exist and are unique up to a dilation and rotation of $R^n$; no rotationally symmetric harmonic diffeomorphism $R^n_*\to H^n_*$ exists). Determine the remaining cases without any symmetry hypothesis; these are the following three questions, each posed separately: (1) for $n\ge 3$, does there exist a harmonic diffeomorphism from $R^n$ onto $H^n$, and does there exist one from $H^n$ onto $R^n$? (2) for $n\ge 3$, does there exist a harmonic diffeomorphism, not necessarily rotationally symmetric, from $R^n_*$ onto $H^n_*$ (the $n=2$ instance is settled, since restricting the stated harmonic diffeomorphism $C\to H^2$ to the punctured spaces yields one)? (3) is every harmonic diffeomorphism $H^n_*\to R^n_*$ ($n\ge 2$) rotationally symmetric up to a similarity of $R^n$, or does a non-rotationally-symmetric harmonic diffeomorphism exist?

References

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