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Journal articles on the topic 'Bergman metrics'

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Published: 6 September 2023

Last updated: 7 September 2023

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1

Herbort, Gregor. "The growth of the bergman kernel on pseudoconvex domains of homogeneous finite diagonal type." Nagoya Mathematical Journal 126 (June 1992): 1–24. http://dx.doi.org/10.1017/s0027763000003986.

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In this article we continue the investigations on invariant metrics on a certain class of weakly pseudoconvex domains which we began in [H 1]. While in that paper the differential metrics of Caratheodory and Kobayashi were estimated precisely, the present paper contains a sharp estimate of the singularity of the Bergman kernel and metric on domains belonging to that class.

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2

Potash, Eric. "Euclidean Embeddings and Riemannian Bergman Metrics." Journal of Geometric Analysis 26, no. 1 (October 26, 2015): 499–528. http://dx.doi.org/10.1007/s12220-015-9560-3.

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3

Feng, Renjie. "Bergman metrics and geodesics in the space of Kähler metrics on principally polarized abelian varieties." Journal of the Institute of Mathematics of Jussieu 11, no. 1 (June 21, 2011): 1–25. http://dx.doi.org/10.1017/s1474748011000119.

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AbstractIt is well known in Kähler geometry that the infinite-dimensional symmetric space $\mathcal{H}$ of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds $\mathcal{B}_k\subset\mathcal{H}$ of Bergman metrics of height k. Then it is natural to ask whether geodesics in $\mathcal{H}$ can be approximated by Bergman geodesics in $\mathcal{B}_k$. For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

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4

LOI, ANDREA. "BERGMAN AND BALANCED METRICS ON COMPLEX MANIFOLDS." International Journal of Geometric Methods in Modern Physics 02, no. 04 (August 2005): 553–61. http://dx.doi.org/10.1142/s0219887805000685.

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5

Ferrari, Frank, Semyon Klevtsov, and Steve Zelditch. "Simple matrix models for random Bergman metrics." Journal of Statistical Mechanics: Theory and Experiment 2012, no. 04 (April 25, 2012): P04012. http://dx.doi.org/10.1088/1742-5468/2012/04/p04012.

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6

Maitani, Fumio, and Hiroshi Yamaguchi. "Variation of Bergman metrics on Riemann surfaces." Mathematische Annalen 330, no. 3 (June 8, 2004): 477–89. http://dx.doi.org/10.1007/s00208-004-0556-8.

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7

BERMAN, ROBERT J. "BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS." International Journal of Mathematics 21, no. 01 (January 2010): 77–115. http://dx.doi.org/10.1142/s0129167x10005933.

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Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

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8

Feng, Zhiming. "The first two coefficients of the Bergman function expansions for Cartan–Hartogs domains." International Journal of Mathematics 29, no. 06 (June 2018): 1850043. http://dx.doi.org/10.1142/s0129167x1850043x.

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Let [Formula: see text] be a globally defined real Kähler potential on a domain [Formula: see text], and [Formula: see text] be a Kähler metric on the Hartogs domain [Formula: see text] associated with the Kähler potential [Formula: see text]. First, we obtain explicit formulas of the coefficients [Formula: see text] of the Bergman function expansion for the Hartogs domain [Formula: see text] in a momentum profile [Formula: see text]. Second, using explicit expressions of [Formula: see text], we obtain necessary and sufficient conditions for the coefficients [Formula: see text] to be constants. Finally, we obtain all the invariant complete Kähler metrics on Cartan–Hartogs domains such that their the coefficients [Formula: see text] of the Bergman function expansions are constants.

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9

Lazaroiu, Calin Iuliu, Daniel McNamee, and Christian Sämann. "Generalized Berezin quantization, Bergman metrics and fuzzy laplacians." Journal of High Energy Physics 2008, no. 09 (September 10, 2008): 059. http://dx.doi.org/10.1088/1126-6708/2008/09/059.

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10

Loi, Andrea, and Fabio Zuddas. "Partially regular and cscK metrics." International Journal of Mathematics 31, no. 10 (July 27, 2020): 2050079. http://dx.doi.org/10.1142/s0129167x20500792.

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A Kähler metric [Formula: see text] with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to [Formula: see text] is a positive constant for all integer [Formula: see text] sufficiently large. The aim of this paper is to prove that for all [Formula: see text] there exists an [Formula: see text]-dimensional complex manifold equipped with strictly partially regular and cscK metric [Formula: see text]. Further, for [Formula: see text], the (constant) scalar curvature of [Formula: see text] can be chosen to be zero, positive or negative.

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11

Song, Jian, and Steve Zelditch. "Bergman metrics and geodesics in the space of Kähler metrics on toric varieties." Analysis & PDE 3, no. 3 (July 21, 2010): 295–358. http://dx.doi.org/10.2140/apde.2010.3.295.

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12

Wulan, Hasi, and Kehe Zhu. "Lipschitz Type Characterizations for Bergman Spaces." Canadian Mathematical Bulletin 52, no. 4 (December 1, 2009): 613–26. http://dx.doi.org/10.4153/cmb-2009-060-6.

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AbstractWe obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.

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13

Hahn, K. T., and P. Pflug. "The Kobayashi and Bergman metrics on generalized Thullen domains." Proceedings of the American Mathematical Society 104, no. 1 (January 1, 1988): 207. http://dx.doi.org/10.1090/s0002-9939-1988-0958068-4.

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14

Lu, Wen, Xiaonan Ma, and George Marinescu. "Optimal convergence speed of Bergman metrics on symplectic manifolds." Journal of Symplectic Geometry 18, no. 4 (2020): 1091–126. http://dx.doi.org/10.4310/jsg.2020.v18.n4.a5.

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15

Fu, Siqi, and Bun Wong. "On strictly pseudoconvex domains with Kähler-Einstein Bergman metrics." Mathematical Research Letters 4, no. 5 (1997): 697–703. http://dx.doi.org/10.4310/mrl.1997.v4.n5.a7.

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16

Cho, Sanghyun, and Young Hwan You. "Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type inC3." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/697160.

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LetΩbe a smoothly bounded pseudoconvex domain inC3and assume thatz0∈bΩis a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curveΓ⊂Ω∪{z0}, connecting points q0∈Ωandz0∈bΩ, and a quantityM(z,X), alongz∈Γ, which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.

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17

OHSAWA, Takeo. "A REMARK ON KAZHDAN'S THEOREM ON SEQUENCES OF BERGMAN METRICS." Kyushu Journal of Mathematics 63, no. 1 (2009): 133–37. http://dx.doi.org/10.2206/kyushujm.63.133.

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18

Coman, Dan, Semyon Klevtsov, and George Marinescu. "Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces." Indiana University Mathematics Journal 68, no. 2 (2019): 593–628. http://dx.doi.org/10.1512/iumj.2019.68.7589.

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19

Herbort, Gregor. "On the invariant differential metrics near pseudoconvex boundary points where the Levi form has corank one." Nagoya Mathematical Journal 130 (June 1993): 25–54. http://dx.doi.org/10.1017/s0027763000004414.

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Let D be a bounded domain in Cn; in the space L2(D) of functions on D which are square-integrable with respect to the Lebesgue measure d2nz the holomorphic functions form a closed subspace H2(D). Therefore there exists a well-defined orthogonal projection PD: L2(D) → H2(D) with an integral kernel KD:D × D → C, the Bergman kernel function of D. An explicit computation of this function directly from the definition is possible only in very few cases, as for instance the unit ball, the complex “ellipsoids” , or the annulus in the plane. Also, there is no hope of getting information about the function KD in the interior of a general domain. Therefore the question for an asymptotic formula for the Bergman kernel near the boundary of D arises.

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20

Hezari, Hamid, Zhiqin Lu, and Hang Xu. "Off-diagonal Asymptotic Properties of Bergman Kernels Associated to Analytic Kähler Potentials." International Mathematics Research Notices 2020, no. 8 (May 7, 2018): 2241–86. http://dx.doi.org/10.1093/imrn/rny081.

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Abstract We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^{-\frac 14}$. These improve the earlier results in the subject for smooth potentials, where an expansion exists in a $k^{-\frac 12}$ neighborhood of the diagonal. We obtain our results by finding upper bounds of the form $C^m m!^{2}$ for the Bergman coefficients $b_m(x, \bar y)$, which is an interesting problem on its own. We find such upper bounds using the method of [3]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results. In the special case of metrics with local constant holomorphic sectional curvatures, we obtain off-diagonal asymptotic in a fixed (as $k \to \infty$) neighborhood of the diagonal, which recovers a result of Berman [2] (see Remark 3.5 of [2] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod $O(e^{-k \delta } )$.

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21

Rubinstein, Yanir A., Gang Tian, and Kewei Zhang. "Basis divisors and balanced metrics." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 778 (April 29, 2021): 171–218. http://dx.doi.org/10.1515/crelle-2021-0017.

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Abstract Using log canonical thresholds and basis divisors Fujita–Odaka introduced purely algebro-geometric invariants δ m {\delta_{m}} whose limit in m is now known to characterize uniform K-stability on a Fano variety. As shown by Blum–Jonsson this carries over to a general polarization, and together with work of Berman, Boucksom, and Jonsson, it is now known that the limit of these δ m {\delta_{m}} -invariants characterizes uniform Ding stability. A basic question since Fujita–Odaka’s work has been to find an analytic interpretation of these invariants. We show that each δ m {\delta_{m}} is the coercivity threshold of a quantized Ding functional on the mth Bergman space and thus characterizes the existence of balanced metrics. This approach has a number of applications. The most basic one is that it provides an alternative way to compute these invariants, which is new even for ℙ n {{\mathbb{P}}^{n}} . Second, it allows us to introduce algebraically defined invariants that characterize the existence of Kähler–Ricci solitons (and the more general g-solitons of Berman–Witt Nyström), as well as coupled versions thereof. Third, it leads to approximation results involving balanced metrics in the presence of automorphisms that extend some results of Donaldson.

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22

Hamano, Sachiko, and Hiroshi Yamaguchi †. "A Note on variation of bergman metrics on riemann surfaces under pseudoconvexity." Complex Variables, Theory and Application: An International Journal 49, no. 7-9 (June 10, 2004): 673–79. http://dx.doi.org/10.1080/02781070412331272523.

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23

Li, Song-Ying, and Ezequias Simon. "Boundary behavior of harmonic functions in metrics of Bergman type on the polydisc." American Journal of Mathematics 124, no. 5 (2002): 1045–57. http://dx.doi.org/10.1353/ajm.2002.0028.

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24

Feng, Zhiming. "On the first two coefficients of the Bergman function expansion for radial metrics." Journal of Geometry and Physics 119 (September 2017): 256–71. http://dx.doi.org/10.1016/j.geomphys.2017.05.007.

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25

Yin, Weiping. "The comparison theorem for the bergman and kobayashi metrics on certain pseudoconvex domains." Complex Variables, Theory and Application: An International Journal 34, no. 4 (December 1997): 351–73. http://dx.doi.org/10.1080/17476939708815059.

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26

Wang, Fang. "On the scattering operators for ACHE metrics of Bergman type on strictly pseudoconvex domains." Advances in Mathematics 309 (March 2017): 306–33. http://dx.doi.org/10.1016/j.aim.2017.01.020.

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27

Rubinstein, Yanir A., and Steve Zelditch. "Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties." Journal of Symplectic Geometry 8, no. 3 (2010): 239–65. http://dx.doi.org/10.4310/jsg.2010.v8.n3.a1.

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28

Li, Song-Ying, and Ezequias Simon. "On Proper Harmonic Maps between Strictly Pseudoconvex Domains with Kahler Metrics of Bergman Type." Asian Journal of Mathematics 11, no. 2 (2007): 251–76. http://dx.doi.org/10.4310/ajm.2007.v11.n2.a5.

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29

Li, Song-Ying, and Lei Ni. "On the holomorphicity of proper harmonic maps between unit balls with the Bergman metrics." Mathematische Annalen 316, no. 2 (February 1, 2000): 333–54. http://dx.doi.org/10.1007/s002080050015.

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30

Zhao, Xiaoxia, Li Ding, and Weiping Yin. "The comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the second type*." Progress in Natural Science 14, no. 2 (February 1, 2004): 105–12. http://dx.doi.org/10.1080/10020070412331343221.

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31

Weiping, Yin, and Zhao Xiaoxia. "The Comparison Theorem for Bergman and Kobayashi Metrics on Cartan-Hartogs Domain of the Third Type." Complex Variables, Theory and Application: An International Journal 47, no. 3 (March 2002): 183–201. http://dx.doi.org/10.1080/02781070290001427.

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32

Kai, Chifune, and Takeo Ohsawa. "A Note on the Bergman metric of Bounded homogeneous Domains." Nagoya Mathematical Journal 186 (2007): 157–63. http://dx.doi.org/10.1017/s0027763000009399.

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AbstractWe show that the Bergman metric of a bounded homogeneous domain has a potential function whose gradient has a constant norm with respect to the Bergman metric, and further that this constant is independent of the choice of such a potential function.

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33

Aryasomayajula, Anilatmaja, and Indranil Biswas. "Bergman kernel on Riemann surfaces and Kähler metric on symmetric products." International Journal of Mathematics 30, no. 14 (October 8, 2019): 1950071. http://dx.doi.org/10.1142/s0129167x1950071x.

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Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

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34

Ruan, Wei-Dong. "Canonical coordinates and Bergmann metrics." Communications in Analysis and Geometry 6, no. 3 (1998): 589–631. http://dx.doi.org/10.4310/cag.1998.v6.n3.a5.

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35

Dong, Robert Xin. "Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves." Complex Manifolds 4, no. 1 (February 23, 2017): 7–15. http://dx.doi.org/10.1515/coma-2017-0002.

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Abstract We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

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36

Herbort, Gregor. "Localization lemmas for the Bergman metric at plurisubharmonic peak points." Nagoya Mathematical Journal 171 (2003): 107–25. http://dx.doi.org/10.1017/s0027763000025538.

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AbstractLet D be a bounded pseudoconvex domain in ℂn and ζ ∈ D. By KD and BD we denote the Bergman kernel and metric of D, respectively. Given a ball B = B(ζ, R), we study the behavior of the ratio KD/KD∩B(w) when w ∈ D ∩ B tends towards ζ. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to 1, as w tends to ζ, under an additional assumption on the pluricomplex Green function D(·, w) of D with pole at w, namely that the diameter of the sublevel sets Aw :={z ∈ D | D(z, w) < −1} tends to zero, as w → ζ. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in [DiOh1], where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set Aw is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).

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37

Krantz, Steven G., and Jiye Yu. "On the Bergman invariant and curvatures of the Bergman metric." Illinois Journal of Mathematics 40, no. 2 (June 1996): 226–44. http://dx.doi.org/10.1215/ijm/1255986102.

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38

CHEN, BO-YONG. "THE BERGMAN METRIC ON TEICHMÜLLER SPACE." International Journal of Mathematics 15, no. 10 (December 2004): 1085–91. http://dx.doi.org/10.1142/s0129167x04002697.

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39

Barletta, Elisabetta, Sorin Dragomir, and Francesco Esposito. "Weighted Bergman Kernels and Mathematical Physics." Axioms 9, no. 2 (April 29, 2020): 48. http://dx.doi.org/10.3390/axioms9020048.

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We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω ⊂ C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form γ = | φ | m on strictly pseudoconvex domains Ω = { φ < 0 } ⊂ C n ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on ∂ Ω × S 1 ). Several open problems are indicated throughout the survey.

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40

Chen, Bo-Yong. "Boundary behavior of the Bergman metric." Nagoya Mathematical Journal 168 (2002): 27–40. http://dx.doi.org/10.1017/s0027763000008333.

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AbstractLet Ω be a bounded pseudoconvex domain in Cn. We give sufficient conditions for the Bergman metric to go to infinity uniformly at some boundary point, which is stated by the existence of a Hölder continuous plurisubharmonic peak function at this point or the verification of property (P) (in the sense of Coman) which is characterized by the pluricomplex Green function.

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41

Diederich, K., and J. E. Fornæss. "Boundary behavior of the Bergman metric." Asian Journal of Mathematics 22, no. 2 (2018): 291–98. http://dx.doi.org/10.4310/ajm.2018.v22.n2.a6.

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42

Herbort, Gregor. "The Bergman metric on hyperconvex domains." Mathematische Zeitschrift 232, no. 1 (September 1999): 183–96. http://dx.doi.org/10.1007/pl00004754.

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43

Chen, Bo-Yong. "Bergman completeness of hyperconvex manifolds." Nagoya Mathematical Journal 175 (2004): 165–70. http://dx.doi.org/10.1017/s0027763000008941.

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44

Choe, Boo Rim, and Heungsu Yi. "Representations and interpolations of harmonic Bergman functions on half-spaces." Nagoya Mathematical Journal 151 (June 1998): 51–89. http://dx.doi.org/10.1017/s0027763000025174.

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Abstract.On the setting of the half-space of the euclidean n-space, we prove representation theorems and interpolation theorems for harmonic Bergman functions in a constructive way. We also consider the harmonic (little) Bloch spaces as limiting spaces. Our results show that well-known phenomena for holomorphic cases continue to hold. Our proofs of representation theorems also yield a uniqueness theorem for harmonic Bergman functions. As an application of interpolation theorems, we give a distance estimate to the harmonic little Bloch space. In the course of the proofs, pseudohyperbolic balls are used as substitutes for Bergman metric balls in the holomorphic case.

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45

Kim, Jong Jin. "A METRIC INDUCED BY THE BERGMAN KERNEL." Honam Mathematical Journal 36, no. 4 (December 25, 2014): 853–62. http://dx.doi.org/10.5831/hmj.2014.36.4.853.

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46

Błocki, Zbigniew. "Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity." Nagoya Mathematical Journal 185 (2007): 143–50. http://dx.doi.org/10.1017/s0027763000025782.

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AbstractFor a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

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47

Dinew, Żywomir. "On the completeness of a metric related to the Bergman metric." Monatshefte für Mathematik 172, no. 3-4 (May 3, 2013): 277–91. http://dx.doi.org/10.1007/s00605-013-0501-6.

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48

Suzuki, Masaaki. "The generalized Schwarz lemma for the Bergman metric." Pacific Journal of Mathematics 117, no. 2 (April 1, 1985): 429–42. http://dx.doi.org/10.2140/pjm.1985.117.429.

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49

Boas, Harold P., Emil J. Straube, and Ji Ye Yu. "Boundary limits of the Bergman kernel and metric." Michigan Mathematical Journal 42, no. 3 (1995): 449–61. http://dx.doi.org/10.1307/mmj/1029005306.

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50

Błocki, Zbigniew. "The Bergman metric and the pluricomplex Green function." Transactions of the American Mathematical Society 357, no. 7 (March 1, 2005): 2613–25. http://dx.doi.org/10.1090/s0002-9947-05-03738-4.

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