Nonvanishing of the Jacobian for harmonic quasiconformal mappings in dimensions $n\ge4$ (Problem 7.3 of Todorčević, 2019)

A mapping between domains in $\mathbb{R}^n$ is harmonic if each coordinate function is harmonic, and $K$-quasiconformal if it is a homeomorphism of bounded distortion, $\|Df(x)\|^n\le K\,J_f(x)$ almost everywhere. In the plane, Lewy's theorem (1936) guarantees that a one-to-one harmonic mapping has nonvanishing Jacobian, so planar harmonic homeomorphisms are diffeomorphisms. In dimensions $n\ge3$ this fails for general harmonic homeomorphisms (Wood 1991), but Wood's examples have unbounded distortion and are not quasiconformal. Problem 7.3 of Todorčević's monograph Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics (2019) asks whether quasiconformality restores the conclusion: does every harmonic quasiconformal mapping between domains in $\mathbb{R}^n$, $n\ge3$, have nonvanishing Jacobian? The question is the identified bottleneck for the co-Lipschitz (hence bi-Lipschitz) property of spatial harmonic quasiconformal maps, the higher-dimensional analogue of Pavlović's planar bi-Lipschitz theorem. Recent work has settled the low-dimensional cases: Kalaj and Zhu (2026, preprint) prove a three-dimensional Lewy theorem — every sense-preserving harmonic quasiregular map in $\mathbb{R}^3$ has strictly positive Jacobian — and Kalaj (2026, preprint) proves a four-dimensional version under the extra assumption of pluriharmonicity. The general case for dimensions $n\ge4$ remains open, including the key blow-up obstruction: whether a homogeneous harmonic polynomial mapping of degree $m>1$ with strictly positive Jacobian on the sphere can exist.

Background

A mapping $f=(f_1,\dots,f_n):\Omega\to\mathbb{R}^n$ on a domain $\Omega\subset\mathbb{R}^n$ is harmonic if each coordinate function satisfies $\Delta f_i=0$; such maps are real analytic. The Jacobian is $J_f=\det Df$. A homeomorphism $f$ is $K$-quasiconformal if it lies in $W^{1,n}_{loc}$ and satisfies the bounded-distortion inequality $\|Df(x)\|^n\le K\,J_f(x)$ almost everywhere, where $\|Df(x)\|$ is the operator norm of the differential; $K=1$ corresponds to conformal maps. Dropping the homeomorphism assumption while keeping the inequality gives the quasiregular maps (sense-preserving means $J_f\ge0$), the higher-dimensional analogue of holomorphic functions; nonconstant quasiregular maps are open and discrete. The intersection — maps that are both harmonic and quasiconformal (HQC) — is a natural but rigid class, and in the plane its theory is highly developed.

The planar theory rests on a theorem of Lewy (1936): a one-to-one harmonic mapping between planar domains has nonvanishing Jacobian, so every planar harmonic homeomorphism is automatically a local diffeomorphism. In dimensions $n\ge3$ this fails: Wood (1991) constructed a harmonic homeomorphism of $\mathbb{R}^3$ whose Jacobian vanishes on an entire plane, with the same construction in every dimension $n\ge3$. Wood's example has unbounded distortion — it is not quasiconformal — so it leaves open the bounded-distortion version of the question: do harmonicity and quasiconformality together force the Jacobian to be nonzero at every point? This is Problem 7.3 of Todorčević's monograph (2019), and the surrounding literature treats it as the central open regularity question for spatial harmonic quasiconformal mappings.

The question is the known bottleneck for the metric theory of the class. Pavlović (2002) proved that harmonic quasiconformal self-maps of the unit disk are bi-Lipschitz, meaning both $|f(x)-f(y)|\le L|x-y|$ and the reverse, co-Lipschitz inequality $|f(x)-f(y)|\ge L^{-1}|x-y|$ hold. In space, the upper Lipschitz bound holds in all dimensions (Astala and Manojlović 2015, with earlier elliptic-regularity proofs, extended to maps between domains with $C^{1,\alpha}$ boundary by Gjokaj and Kalaj 2026), but the co-Lipschitz bound is blocked exactly by the Jacobian question: a co-Lipschitz map has nonvanishing Jacobian everywhere, and the inverse of a harmonic map is generally not harmonic, so co-Lipschitz cannot be read off from quasiconformality of the inverse. Astala and Manojlović obtained bi-Lipschitz regularity in the ball only under the extra assumption that $f=\nabla u$ for a harmonic function $u$, and Božin and Mateljević obtained co-Lipschitz conclusions only under small distortion ($K<3^{n-1}$) onto convex targets.

Very recent work has transformed the low-dimensional picture. Kalaj and Zhu (2026, preprint) proved a three-dimensional Lewy theorem: every nonconstant sense-preserving harmonic quasiregular map $\Omega\subset\mathbb{R}^3\to\mathbb{R}^3$ has $J_f>0$ everywhere, so harmonic quasiconformal maps in three-space are real-analytic diffeomorphisms. Their proof blows up a hypothetical zero of $J_f$ to a homogeneous harmonic polynomial quasiregular map and excludes it through a second-order trace identity that contradicts the maximum principle on the sphere $S^2$. They also show precisely why the method stops there: for $n\ge4$ the quadratic term of the trace identity becomes indefinite, and in even dimensions there are explicit homogeneous harmonic cubic maps with $J\ge0$ on the sphere, sitting on the borderline. Kalaj (2026, preprint) proved a four-dimensional Lewy theorem only under the additional complex-analytic assumption that the map is pluriharmonic (each complex component locally the real part of a holomorphic function). Consequently the general question in dimensions $n\ge4$ remains open.

Problem Statement

Let $n\ge4$, and let $f:\Omega\to\Omega'$ be a harmonic quasiconformal homeomorphism between domains in $\mathbb{R}^n$: each coordinate of $f$ is harmonic in $\Omega$, and $f$ is a $K$-quasiconformal homeomorphism for some finite $K$ ($\|Df(x)\|^n\le K\,J_f(x)$ almost everywhere). Must $J_f(x)=\det Df(x)$ be nonzero at every point $x\in\Omega$? Equivalently, is every harmonic quasiconformal map between domains in $\mathbb{R}^n$, $n\ge4$, a local $C^\infty$-diffeomorphism?

This is the surviving core of Problem 7.3 of Todorčević (2019), which posed the question for all $n\ge3$: the case $n=2$ is Lewy's classical theorem, and the case $n=3$ was proved affirmatively by Kalaj and Zhu (2026, preprint) in the stronger form that every sense-preserving harmonic quasiregular map in $\mathbb{R}^3$ has strictly positive Jacobian. In the surviving case $n\ge4$ the question is universal in both the map and the dimension — whether $J_f\ne0$ everywhere for every harmonic quasiconformal homeomorphism between domains in $\mathbb{R}^n$ and every $n\ge4$ — so that a single such homeomorphism in some $\mathbb{R}^n$, $n\ge4$, with vanishing Jacobian at an interior point would contradict the universal claim; the analogous assertion for sense-preserving harmonic quasiregular maps is stronger. A closely tied sub-obstruction, posed by Kalaj and Zhu, is the homogeneous version: does there exist, for some $n\ge4$, a homogeneous harmonic polynomial mapping $P:\mathbb{R}^n\to\mathbb{R}^n$ of degree $m>1$ with $J_P>0$ on the unit sphere $S^{n-1}$? A negative answer to this homogeneous question — ruling out every such $P$ for every $n\ge4$ — yields the affirmative answer to the main question.

References

1. Vesna Todorčević, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, Springer (Birkhäuser), Cham, 2019. DOI: 10.1007/978-3-030-22591-9. https://doi.org/10.1007/978-3-030-22591-9
2. Hans Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bulletin of the American Mathematical Society 42 (1936), 689–692. DOI: 10.1090/s0002-9904-1936-06397-4. https://doi.org/10.1090/s0002-9904-1936-06397-4
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7. David Kalaj and Jian-Feng Zhu, A Lewy theorem for harmonic quasiregular mappings in three-space, arXiv preprint arXiv:2607.04720 (v3, 2026). https://arxiv.org/abs/2607.04720
8. David Kalaj, A Lewy-type theorem for pluriharmonic mappings in $\mathbb{C}^2$ and bi-Lipschitz quasiconformal harmonic maps, arXiv preprint arXiv:2608.11948 (v2, 2026). https://arxiv.org/abs/2608.11948
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