Additive Riemann–Hilbert Problem in Line Bundles Over ℂℙ1

2006 ◽  
Vol 49 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Roman J. Dwilewicz
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Hilbert Problem ◽  
Holomorphic Functions ◽  
Riemann Function ◽  
Line Bundles ◽  
Complex Projective ◽  
Riemann Hilbert Problem

AbstractIn this note we consider -problem in line bundles over complex projective space ℂℙ1 and prove that the equation can be solved for (0, 1) forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to ℂℙ2 since by removing a point from it we get a line bundle over ℂℙ1.

scholarly journals A MINIMAL REAL HYPERSURFACE OF A COMPLEX PROJECTIVE SPACE WITH NONNEGATIVE SECTIONAL CURVATURE

2010 ◽  
Vol 81 (3) ◽  
pp. 488-492
Author(s):  
MAYUKO KON
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Nonnegative Sectional Curvature ◽  
Complex Projective

AbstractWe give a characterization of a minimal real hypersurface with respect to the condition for the sectional curvature.

scholarly journals Real hypersurfaces of a complex projective space II

1984 ◽  
Vol 30 (1) ◽  
pp. 123-127 ◽  
Author(s):  
Sadahiro Maeda
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Real Hypersurfaces ◽  
Complex Projective

We consider a certain real hypersurface M of a complex projective space. The purpose of this paper is to characterize M in terms of Ricci curvatures.

scholarly journals The index of harmonic maps from surfaces to complex projective spaces

2020 ◽  
Vol 31 (09) ◽  
pp. 2050069
Author(s):  
J. Oliver
Keyword(s):  
Projective Space ◽  
Lower Bounds ◽  
Complex Projective Space ◽  
Harmonic Maps ◽  
Projective Spaces ◽  
Line Bundles ◽  
Complex Projective Spaces ◽  
Holomorphic Sections ◽  
Higher Genus ◽  
Complex Projective

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.

scholarly journals REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

2013 ◽  
Vol 24 (07) ◽  
pp. 1350050 ◽  
Author(s):  
JURGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Kähler Manifolds ◽  
Real Hypersurfaces ◽  
Kahler Manifolds ◽  
Totally Geodesic ◽  
Open Part ◽  
Complex Submanifold ◽  
Complex Projective

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

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