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Two boundary rigidity results for holomorphic maps. (English) Zbl 1495.32041

The author establishes two boundary rigidity results for holomorphic mappings that give substantial improvements over existing results.
The first is one is the following.
Theorem 1.5. Suppose \(\Omega\subset \mathbb{C}^{d}\) is a bounded convex domain with \(C^{2}\) boundary. If \(f:\Omega\rightarrow\Omega\) is a holomorphic map and there exists \(\xi_{0}\in \partial\Omega\) such that \(f(z) = z + \mathcal{O}(\|z-\xi_{0}\|^{4})\), then \(f = \operatorname{id}\).
The main ingredients in the proof are a study of complex geodesics and their images under \(f\) and recent estimates on the Kobayashi distance due to [A. Christodoulou and I. Short, Ann. Acad. Sci. Fenn., Math. 44, No. 1, 293–300 (2019; Zbl 1423.30032)]. In previous results, the domains were assumed to be of finite type, and the exponent in the \(\mathcal{O}\) term depended on the type (which in the case of convex domains coincides with the line type); compare also the Greene-Krantz conjecture.
The second boundary rigidity result reads as follows.
Theorem 1.13. Suppose \(\Omega\subset\mathbb{C}^{d}\) is a bounded domain, \(\varphi\in \operatorname{Aut}(\Omega)\), \(\partial\Omega\) satisfies an interior cone condition at \(\xi_{0}\in \partial\Omega\) with parameter \(\theta\), and there exists a \(\varphi\)-invariant Kähler metric \(g\) on \(\Omega\) with property-(BG) with parameters \(\kappa\), \(A\). If \[L>4d+2+\frac{\sqrt{\kappa}A}{\sin(\theta)} \] and \[\varphi(z) = z + \mathcal{O}(\|z-\xi_{0}\|^{L})\;, \] then \(\varphi = \operatorname{id}\).
Here \(\theta\) is the aperture of the cone in the cone condition. A complete Kähler metric \(g\) has property (BG) (for bounded geometry) if the following two conditions hold:
1)
the sectional curvature of \(g\) is bounded in absolute value by \(\kappa>0\),
2)
there is \(A>0\) such that \(\sqrt{g_{z}(v,v)}\leq A\frac{\|v\|}{\delta_{\Omega}(z)}\) for all \(z\in\Omega\), \(v\in\mathbb{C}^{d}\).
Here, \(\delta_{\Omega}(z) = \inf\{\|w-z\|:w\in \partial\Omega\}\). The author’s approach is new, in that it makes no assumptions on the CR-geometry of the boundary, but instead makes assumptions about the intrinsic geometry of the domain.

MSC:

32H12 Boundary uniqueness of mappings in several complex variables
32Q15 Kähler manifolds

Citations:

Zbl 1423.30032
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References:

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