Analysis of the Wu metric II: The case of non-convex Thullen domains

Cheung, Chi-Keung; Kim, K.

doi:10.1090/s0002-9939-97-03695-2

Cited by 10 publications

(10 citation statements)

References 23 publications

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“…This result extends the work of Cheung and Kim ([3] and [4]) on pseudo-egg domains in C 2 . It also verifies the following conjecture of Kobayashi ([10]) (refer [13] for a modified version of this…”

Section: Introductionsupporting

confidence: 89%

“…To establish the strong negativity of the holomorphic curvature near the origin, in the sense of currents, compare the restriction G = i * g | S with H = h E 2m | S . Real analyticity of G ensures that its logarithmic average on small 'discs' about s = 0 in S equals ∆ log G(s) at s = 0 -this can be verified by a Taylor expansion of G in s, s. The decreasing property of holomorphic curvature together with the facts that G is a conformal metric on S of constant negative holomorphic curvature and H ≥ 1/ √ nG, establishes the strong negativity of the holomorphic curvature current of the Wu metric in a neighbourhood of the origin as in [4]. Notice that the bound on the curvature that we get here, is a constant that depends on the dimension n. But this constant does not depend on m. As Z is the orbit of the origin under the action of Aut(E 2m ), this analysis carries forth to hold throughout Z.…”

Section: Proposition 32mentioning

confidence: 75%

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Analysing the Wu metric on a class of eggs in ℂ n – II

Balakumar

Mahajan

2017

Proc Math Sci

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We study the Wu metric for the non-convex domains of the formwhere 0 < m < 1/2. Explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs E2m are obtained. The Wu metric is then verified to be a continuous Hermitian metric on E2m which is real analytic everywhere except along the complex hypersurface Z = {(0, z2, . . . , zn) ∈ E2m}. We also show that the holomorphic sectional curvature of the Wu metric for this non-compact family of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of m. This verifies a conjecture of S. Kobayashi and H. Wu for such E2m.1991 Mathematics Subject Classification. Primary: 32F45; Secondary: 32Q45, 32H15.

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Section: Introductionsupporting

confidence: 89%

Section: Proposition 32mentioning

confidence: 75%

Analysing the Wu metric on a class of eggs in ℂ n – II

Balakumar

Mahajan

2017

Proc Math Sci

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“…, where m > 0, was obtained by Cheung and Kim in [5] and [6]. This is a natural example arising in the pursuit of well-adapted metrics in complex analysis, the background for which can be found in [13] and [9].…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this note and its counterpart [2], is to carry forward the case study of the Wu metric in conjunction with (K-W) as in [5] and [6], to egg domains of the form (1.1)…”

Section: Introductionmentioning

confidence: 99%

Analysing the Wu metric on a class of eggs in ℂ n – I

Balakumar

Mahajan

2017

Proc Math Sci

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We study the Wu metric on convex egg domains of the form E2m = z ∈ C n : |z1| 2m + |z2| 2 + . . . + |zn−1| 2 + |zn| 2 < 1 where m ≥ 1/2, m = 1. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be C 2 -smooth. Overall however, the Wu metric is shown to be continuous when m = 1/2 and even C 1 -smooth for each m > 1/2, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such E2m.1991 Mathematics Subject Classification. Primary: 32F45; Secondary: 32Q45. Key words and phrases. Wu metric, Kobayashi metric, negative holomorphic curvature. Both the authors were supported by the DST-INSPIRE Fellowship of the Government of India. 1 The term 'holomorphic curvature' stands precisely for the holomorphic sectional curvature and is said to be strongly negative, if it is bounded above by a negative constant.2 We shall reserve the term 'metric' for what was termed as a sub-metric in [13] and use the word 'distance'for what is termed as a 'metric' in general topology and metric geometry. 3 These are also referred to as 'complex ellipsoids' in the complex analysis literature. However, since the ellipsoid in Wu's construction is defined by a polynomial of degree two, we adopt a different terminology.

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“…3 Recent developments in metallocene catalyst technology facilitate the copolymerization of ethylene with relatively large amounts of styrene. 4,5 These poly(ethylene-ran-styrene) copolymers represent a new family of materials that are being used in a variety of applications such as calendaring, filled systems, injection molding and blends. 6,7 In this communication, we report the synthesis and properties of thermoplastic elastomers derived from the ES copolymers, namely lightly sulfonated poly(ethylene-ran-styrene) ionomers.…”

Section: Introductionmentioning

confidence: 99%

Sulfonated poly(ethylene‐ran‐styrene) ionomers

Shim

Weiss

2005

Polymer International

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Ionomers were prepared by partially sulfonating the styrene units of a poly(ethylene-ranstyrene) (ES) copolymer. The metal salt derivatives of sulfonated ES had higher T g , melt viscosity and rubbery modulus than did the parent ES because of the formation of labile, physical crosslinks from intermolecular dipole-dipole associations between metal sulfonate groups. Microphase separation of ionrich aggregates with a characteristic size of 3-9 nm also occurred as a result of the strong intermolecular associations.

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Analysis of the Wu metric II: The case of non-convex Thullen domains

Cited by 10 publications

References 23 publications

Analysing the Wu metric on a class of eggs in ℂ n – II

Analysing the Wu metric on a class of eggs in ℂ n – II

Analysing the Wu metric on a class of eggs in ℂ n – I

Sulfonated poly(ethylene‐ran‐styrene) ionomers

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