For every n≥ 3, let 0<r<1 and let h:A(r,1)→A(R,1)⊂R^n be an onto homeomorphism whose coordinate functions are harmonic, where A(a,b)=x∈R^n:a<|x|<b. If h preserves the two ends of the annulus, then R≤ R_n,+(r):=fracnrn-1+r^n. If h interchanges the two ends, then R≤ R_n,-(r):=fracnr^n-11+(n-1)r^n. Moreover, equality in either bound occurs only, up to an orthogonal transformation, for the corresponding end-preserving or end-reversing radial harmonic homeomorphism.

Higher-Dimensional Nitsche Conjecture

For every $n\ge 3$ , let $0<r<1$ and let $h:\mathbb{A}(r,1)\to\mathbb{A}(R,1)\subset\mathbb{R}^n$ be an onto homeomorphism whose coordinate functions are harmonic, where $\mathbb{A}(a,b)=\{x\in\mathbb{R}^n:a<|x|<b\}$ . If $h$ preserves the two ends of the annulus, then $R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}$ . If $h$ interchanges the two ends, then $R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}$ . Moreover, equality in either bound occurs only, up to an orthogonal transformation, for the corresponding end-preserving or end-reversing radial harmonic homeomorphism.

References

Primary source

arXiv

Additional references

The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to settle the higher-dimensional version of the conjecture, but the result has not yet been independently checked.

The conjecture concerns sharp distortion bounds for annular homeomorphisms in dimensions $n \ge 3$ , including equality cases. Nitsche posed the classical annular conjecture in 1962; the present claim addresses the higher-dimensional setting.

Known results

Nitsche, 1962: posed the classical annular conjecture.
Iwaniec, Kovalev, and Onninen, 2011: proved the classical planar conjecture.
Later work records the final conjectured inequality for $n \ge 3$ as open, including higher-dimensional energy generalizations.

August 27, 2026 claimed resolution

A preprint titled The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity claims both endpoint bounds without boundary-regularity assumptions and classifies equality cases in the stated $n \ge 3$ annular homeomorphism setting. It therefore claims to resolve the sharp question, but the result has not yet been peer reviewed or independently verified.

Current status (as of August 2026): A preprint claims the $n \ge 3$ problem is solved with sharp bounds and rigidity, but that claim remains unverified; the earlier literature recorded the higher-dimensional inequality as open.

Sources

No solutions have been posted yet.