Acta mathematica scientia, Series B >
THE CONJUGATE POINTS OF CP∞ AND THE ZEROES OF BERGMAN KERNEL
Received date: 2008-06-16
Revised date: 2009-01-13
Online published: 2009-05-20
Supported by
Partially support by NSF of China (A01010501 and 10731080)
Two points of the infinite dimensional complex projective space CP∞ with homogeneous coordinates α=(α0, α1, α2, … ) and b=b0, b1, b2, … ), respectively, are conjugate if and only if they are complex orthogonal, i.e., αb = Σ∞j=0
αj bj =0. For a complete ortho-normal system φ(t)=(φ0(t), φ1(t), φ2(t), … ) of L2H(D), the space of the holomorphic and absolutely square integrable functions in the bounded domain D of Cn, φ(t), t ∈ D, is considered as the homogeneous coordinate of a point in CP∞. The correspondence t → φ(t) induces a holomorphic imbedding tφ : D → CP∞. It is proved that the Bergman kernel K(t, v) of D equals to zero for the two points t and v in D if and only if their image points under tφ are conjugate points of CP∞.
Key words： conjugate points; Bergman kernel
Lu Qikeng . THE CONJUGATE POINTS OF CP∞ AND THE ZEROES OF BERGMAN KERNEL[J]. Acta mathematica scientia, Series B, 2009 , 29(3) : 480 -492 . DOI: 10.1016/S0252-9602(09)60048-5
[1] Lu Qikeng. The Poisson formula for harmonic (1.0)-forms in a ball of Cn. Acta Math Sci, 1991, 11: 267--273
[2] Lu Qikeng. A note on self-dual and anti-self-dual gauge field. J Systems Sci & Compl, 1999, 12(Suppl): 70--73
[3] Lu Qikeng. The gauge field and the connection of a principal fibre bundle (in Chinese). Acta Phys Sin, 1974, 23: 249--263
[4] Ye Fangcao. The conjugate points on Grassmann manifold (in Chinese). Acta Math Sinica, 1978, 21: 367--374
[5] Liu Weiming. The conjugate points of the extended spaces II, III and IV (in Chinese)
[D]. Hefei: University of Science and Technology of China, 1990
[6] Lu Qikeng, Xu Yizhao. A note on bounded transitive domain (in Chinese). Acta Math Sin, 1961, 11: 11--23
[7] Bergman S. Sur les fontions orthognales de plusieurs variables complexe avec les applicationàthèories de fonctions analytiques. Paris: Gauthiers-Villags, 1947
[8] Kobayashi S. Geometry of bounded domains. Trans Amer Math Soc, 1959, 92: 267--290
[9] Lu Qikeng. Introduction to the Theory of Functions of Several Complex Variables (in Chinese). Beijing: Science Press, 1962
[10] Skwarczy'nski Macief. The invariant distance in the theory of pseudo-conformal transformatios and the Lu Qi-Keng conjecture. Proc Amer Math Soc, 1969, 22: 305--310
[11] Skwarczy'nski M. Biholomorphic invariants related to the Bergman functions. Disertationes Mathematicae, 173, Poliska Akademmia Nauk, Institut Matematyczny, Warszawa, 1980
/
| 〈 |
|
〉 |