Robin functions for complex manifolds and applications by Kang-Tae Kim

By Kang-Tae Kim

In a prior Memoirs (Vol. ninety two, No. 448), Levenberg and Yamaguchi analysed the second one version of the Robin functionality $-\lambda(t)$ linked to a tender edition of domain names in $\mathbb{C}^n$ for $n\geq 2$. within the present paintings, the authors examine a generalisation of this moment version formulation to complicated manifolds $M$ built with a Hermitian metric $ds^2$ and a gentle, nonnegative functionality $c

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2. and 3. 1. Since dim H = dim g0 ≥ 1, H satisfies 1. Fix ζ ∈ D. We consider the level set Sζ := {z ∈ D : Λ(z) = Λ(ζ)}. Since −Λ is an exhaustion function for D, we have Sζ Λ(ζ exp tX) = Λ(ζ), D. Fix X ∈ g0 . We have for all t ∈ C so that ζ exp tX ⊂ Sζ . 10) D. The conclusion that we have Λ(ζg) = Λ(ζ) for all g ∈ H. Hence ζH ⊂ Sζ that D has a foliation D = ∪z∈D zH follows from this observation together with the fact that M itself is foliated by cosets M = ∪z∈M zH (the second item of the Frobenius theorem).

Then gh−1 ∈ D (z)h−1 = D (h(z)). We take {gn }n=1,2,... ⊂ D (z) with gn → g in G as n → ∞. Since gn h−1 ∈ D (z)h−1 = D (h(z)), we have gh−1 ∈ ∂D (h(z)). From II. we see that for z, z ∈ M , D(z) and D(z ) are biholomorphically equivalent. In case z, z ∈ D, from III. and 1. 1, D (z) and D (z ) are also biholomorphically equivalent. 1. Let ζ ∈ ∂D and let {zn }n ⊂ D converge to ζ as n → ∞. Then there exist gn ∈ ∂D (zn ), n = 1, 2, . , such that gn → e in G as n → ∞. Proof. We take a holomorphic section σ : V → G of the fiber space ψζ : G → M over a neighborhood V ⊂ M of ζ with σ(ζ) = e.

It follows that any smoothly bounded pseudoconvex domain D M is Stein. 2. Grauert gave an example of a pseudoconvex domain D with smooth boundary which is not Stein. Moreover, D admits no nonconstant holomorphic functions. , [5] and Example 3 in p. 324 in [15]). Our goal in the rest of this section is to describe all pseudoconvex subdomains D of T with smooth boundary which are not Stein. The verification will be postponed to Appendix C. 2. We begin with real 4-dimensional Euclidean space R4 with coordinates x = (x1 , x2 , x3 , x4 ).

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