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L-p harmonic 1-forms on totally real submanifolds in a complex projective space

Authors
Choi, HagyunSeo, Keomkyo
Issue Date
Apr-2020
Publisher
SPRINGER
Keywords
Isotropic lift; Totally real submanifold; Complex projective space; Harmonic form
Citation
ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, v.57, no.3, pp.383 - 400
Journal Title
ANNALS OF GLOBAL ANALYSIS AND GEOMETRY
Volume
57
Number
3
Start Page
383
End Page
400
URI
https://scholarworks.sookmyung.ac.kr/handle/2020.sw.sookmyung/2470
DOI
10.1007/s10455-020-09705-w
ISSN
0232-704X
Abstract
Let pi : S2n+1 -> CPn be the Hopf map and let phi be a totally real immersion of a k(>= 3)-dimensional simply connected manifold Sigma into CPn. It is well known that there exists an isotropic lift (phi) over bar into S2n+1 preserving the second fundamental form. Using this isotropic lift, we obtain a vanishing theorem for of L-p harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space provided the L-k norm of the traceless second fundamental form phi is sufficiently small. Moreover, we prove that if the L-k norm of phi is finite, then the dimension of L-p harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space is finite. As consequences, we obtain a vanishing theorem and a finiteness result for L-2 harmonic 1-forms on a complete noncompact minimal Lagrangian submanifold in a complex projective space.
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