Let Omega ⊂ C^d be a pseudoconvex domain of finite type and let xi_0 ∈ partial Omega. A holomorphic self-map is a map f:Omega→Omega that is holomorphic. Burns–Krantz conjecture. There exists an integer m, depending on the geometry of partial Omega at xi_0, such that if f(z)=z+o(|z-xi_0|^m), then f=operatornameid. This asks whether finite type supplies a boundary Schwarz lemma for weakly pseudoconvex domains, extending the known strongly pseudoconvex result; the optimal dependence of m on the boundary geometry is not specified.

Burns–Krantz conjecture for finite-type pseudoconvex domains

About 8 years old · traced to 2018

Let $\Omega \subset \mathbb{C}^d$ be a pseudoconvex domain of finite type and let $\xi_0 \in \partial \Omega$ . A holomorphic self-map is a map $f:\Omega\to\Omega$ that is holomorphic. Burns–Krantz conjecture. There exists an integer $m$ , depending on the geometry of $\partial\Omega$ at $\xi_0$ , such that if

f(z)=z+o\left(\left\|z-\xi_0\right\|^m\right),

then $f=\operatorname{id}$ . This asks whether finite type supplies a boundary Schwarz lemma for weakly pseudoconvex domains, extending the known strongly pseudoconvex result; the optimal dependence of $m$ on the boundary geometry is not specified.

References

Primary source

Andrew Zimmer, “Two boundary rigidity results for holomorphic maps”, arXiv:1810.05669 (2018).

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Open

The full conjecture remains open, with results only for narrower classes such as convex domains.

Attributed to Burns and Krantz, the conjecture asks whether sufficiently high-order boundary agreement forces a holomorphic self-map of a finite-type pseudoconvex domain to be the identity. No proof or counterexample for the full class was found.

Known results

Huang proved the assertion for bounded convex finite-type domains when $m>5\ell(\xi_0)$ , where $\ell(\xi_0)$ is the line type.
For bounded convex domains with $C^2$ boundary, the assertion holds with $m=4$ , without assuming finite type.
In that convex $C^2$ setting, whether $m=4$ can be reduced to $m=3$ was reported as unclear.

Current status (as of September 2026): The full finite-type pseudoconvex conjecture remains open; convex finite-type cases are known, but no result here settles arbitrary finite-type pseudoconvex domains.

Sources

No solutions have been posted yet.