Rigidity of Homogeneous Holomorphic S2 in a Complex Grassmann Manifold G(2, N)

Jie Fei; Ling He; Jun Wang

doi:10.1007/s12220-023-01387-7

Rigidity of Homogeneous Holomorphic S² in a Complex Grassmann Manifold G(2, N)

Jie Fei, Ling He, Jun Wang^*

^*Corresponding author for this work

Xi'an Jiaotong-Liverpool University

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

In this paper, we give a local rigid characterization of all homogeneous holomorphic two-spheres in a complex Grassmann manifold G(2, N). Let κ denote the new global invariant defined in terms of the square norm of (1, 0) part of the second-order covariant differential of the first ∂ -transform for a holomorphic curve in G(2, N). It is proved that a linearly full non-degenerate holomorphic curve of constant curvature and constant square norm of the second fundamental form in G(2, N) with κ= 0 must be homogeneous.

Original language	English
Article number	324
Journal	Journal of Geometric Analysis
Volume	33
Issue number	10
DOIs	https://doi.org/10.1007/s12220-023-01387-7
Publication status	Published - Oct 2023

Keywords

Complex Grassmann manifolds
Gaussian curvature
Holomorphic curves
Rigidity
The second fundamental form

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10.1007/s12220-023-01387-7

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Fei, J., He, L., & Wang, J. (2023). Rigidity of Homogeneous Holomorphic S² in a Complex Grassmann Manifold G(2, N). Journal of Geometric Analysis, 33(10), Article 324. https://doi.org/10.1007/s12220-023-01387-7

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title = "Rigidity of Homogeneous Holomorphic S2 in a Complex Grassmann Manifold G(2, N)",

abstract = "In this paper, we give a local rigid characterization of all homogeneous holomorphic two-spheres in a complex Grassmann manifold G(2, N). Let κ denote the new global invariant defined in terms of the square norm of (1, 0) part of the second-order covariant differential of the first ∂ -transform for a holomorphic curve in G(2, N). It is proved that a linearly full non-degenerate holomorphic curve of constant curvature and constant square norm of the second fundamental form in G(2, N) with κ= 0 must be homogeneous.",

keywords = "Complex Grassmann manifolds, Gaussian curvature, Holomorphic curves, Rigidity, The second fundamental form",

author = "Jie Fei and Ling He and Jun Wang",

note = "Funding Information: The authors would like to express gratitude for the referee{\textquoteright}s valuable comments and suggestions. This work was supported by National Key R &D Program of China (Grant No. 2022YFA1006600) and NSFC (Grant Nos. 11401481, 12071338, 12071352, 11301273, 11971237). The first named author was also supported by the Research Enhancement Fund of Xi{\textquoteright}an Jiaotong-Liverpool University (REF-18-01-03). The third named author was also supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20221320). Publisher Copyright: {\textcopyright} 2023, Mathematica Josephina, Inc.",

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doi = "10.1007/s12220-023-01387-7",

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AU - Fei, Jie

AU - He, Ling

AU - Wang, Jun

N1 - Funding Information: The authors would like to express gratitude for the referee’s valuable comments and suggestions. This work was supported by National Key R &D Program of China (Grant No. 2022YFA1006600) and NSFC (Grant Nos. 11401481, 12071338, 12071352, 11301273, 11971237). The first named author was also supported by the Research Enhancement Fund of Xi’an Jiaotong-Liverpool University (REF-18-01-03). The third named author was also supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20221320). Publisher Copyright: © 2023, Mathematica Josephina, Inc.

PY - 2023/10

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N2 - In this paper, we give a local rigid characterization of all homogeneous holomorphic two-spheres in a complex Grassmann manifold G(2, N). Let κ denote the new global invariant defined in terms of the square norm of (1, 0) part of the second-order covariant differential of the first ∂ -transform for a holomorphic curve in G(2, N). It is proved that a linearly full non-degenerate holomorphic curve of constant curvature and constant square norm of the second fundamental form in G(2, N) with κ= 0 must be homogeneous.

AB - In this paper, we give a local rigid characterization of all homogeneous holomorphic two-spheres in a complex Grassmann manifold G(2, N). Let κ denote the new global invariant defined in terms of the square norm of (1, 0) part of the second-order covariant differential of the first ∂ -transform for a holomorphic curve in G(2, N). It is proved that a linearly full non-degenerate holomorphic curve of constant curvature and constant square norm of the second fundamental form in G(2, N) with κ= 0 must be homogeneous.

KW - Complex Grassmann manifolds

KW - Gaussian curvature

KW - Holomorphic curves

KW - Rigidity

KW - The second fundamental form

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JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 10

M1 - 324

ER -

Rigidity of Homogeneous Holomorphic S² in a Complex Grassmann Manifold G(2, N)

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