Two applications of the Schwarz lemma

Liu, Bingyuan

doi:10.2140/pjm.2018.296.141

Cited by 6 publications

(4 citation statements)

References 10 publications

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“…In recent years, the Schwarz lemma has seen significant advancements in the realm of complex variables and geometry. In 2014, Liu Bingyuan presented dual applications of Schwarz lemmas, emphasizing the connection between domain geometries and curvatures [4]. By 2019, Ni Lei bridged earlier Schwarz Lemmata with his own research, offering insights into Kähler manifolds and their curvatures [5].…”

Section: Introductionmentioning

confidence: 99%

Schwarz's lemma on the perplex plane

Huang,

Zhang

2024

HSET

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Perplex number is an extension of complex numbers, constituting a commutative ring with zero divisors. In this paper, we primarily investigate the Schwarz lemma on the perplex plane and successfully obtain an estimation of the modulus of differentiable functions in perplex plane. Furthermore, we discuss its potential applications in complex analysis and geometric function theory. This conclusion not only offers a fresh perspective for mathematical research but also holds significant reference value for advancing studies in physics, providing valuable theoretical support for subsequent scientific investigations.

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Section: Introductionmentioning

confidence: 99%

Schwarz's lemma on the perplex plane

Huang,

Zhang

2024

HSET

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“…Behrens' proof in the case when M is hyperbolic crucially used Pinchuk's scaling techniques and the fact that the model domain at a strongly pseudoconvex boundary point is the ball. Related work on the union problem can be found in [6] and [24].…”

Section: Introductionmentioning

confidence: 99%

Limits of an increasing sequence of complex manifolds

Balakumar

Borah

Mahajan

et al. 2022

Annali di Matematica

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Let M be a complex manifold which admits an exhaustion by open subsets M j each of which is biholomorphic to a fixed domain ⊂ C n . The main question addressed here is to describe M in terms of . Building on work of Fornaess-Sibony, we study two cases, namely M is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When M is Kobayashi hyperbolic, its complete description is obtained when is one of the following domains-(i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in C 2 , or (iv) a simply connected domain in C 2 with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when is the minimal ball or the symmetrized polydisc in C n can also be handled. When the Kobayashi metric on M has corank one and is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre C over a holomorphic retract of or that of a limiting domain associated with it. Finally, when = × B n−1 , the product of the unit disc ⊂ C and the unit ball B n−1 ⊂ C n−1 , a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and = ×B n−1 , it is shown that M is biholomorphic to . Further, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to Z ×C, where Z ⊂ = ×B n−1 is a holomorphic retract.

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Section: Introductionmentioning

confidence: 99%

Limits of an increasing sequence of complex manifolds

Balakumar¹,

Borah²,

Mahajan³

et al. 2021

Preprint

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Let M be a complex manifold which admits an exhaustion by open subsets Mj each of which is biholomorphic to a fixed domain Ω ⊂ C n . The main question addressed here is to describe M in terms of Ω. Building on work of Fornaess-Sibony, we study two cases namely, M is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When M is Kobayashi hyperbolic, its complete description is obtained when Ω is one of the following domains -(i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in C 2 , or (iv) a simply connected domain in C 2 with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when Ω is the minimal ball or the symmetrized polydisc in C n can also be handled. When the Kobayashi metric on M has corank one and Ω is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre C over a holomorphic retract of Ω or that of a limiting domain associated with it. Finally, when Ω = ∆ × B n−1 , the product of the unit disc ∆ ⊂ C and the unit ballFurther, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to Z × C, where Z ⊂ Ω = ∆ × B n−1 is a holomorphic retract.

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Two applications of the Schwarz lemma

Cited by 6 publications

References 10 publications

Schwarz's lemma on the perplex plane

Schwarz's lemma on the perplex plane

Limits of an increasing sequence of complex manifolds

Limits of an increasing sequence of complex manifolds

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