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Статті в журналах з теми "Holomorphic immersion"

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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Nomizu, Katsumi, Ulrich Pinkall, and Fabio Podestà. "On the geometry of affine Kähler immersions." Nagoya Mathematical Journal 120 (December 1990): 205–22. http://dx.doi.org/10.1017/s0027763000003342.

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Анотація:

In this paper we extend the work on affine immersions [N-Pi]-1 to the case of affine immersions between complex manifolds and lay the foundation for the geometry of affine Kähler immersions. The notion of affine Kähler immersion extends that of a holomorphic and isometric immersion between Kähler manifolds and can be contrasted to the notion of holomorphic affine immersion which has been established in the work of Dillen, Vrancken and Verstraelen [D-V-V] and that of Abe [A].

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2

Nomizu, Katsumi, and Fabio Podestà. "On the Cartan-Norden theorem for affine Kähler immersions." Nagoya Mathematical Journal 121 (March 1991): 127–35. http://dx.doi.org/10.1017/s002776300000341x.

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In [N-Pi-Po] the notion of affine Kähler immersion for complex manifolds has been introduced: if Mn is an n-dimensional complex manifold and f: Mn -→ Cn+1 is a holomorphic immersion together with an anti-holomorphic transversal vector field ζ, we can induce a connection ▽ on Mn which is Kähler-like, that is, its curvature tensor R satisfies R(Z, W) = 0 as long as Z, W are (1, 0) complex vector fields on M.

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3

Alghanemi, Azeb, Noura Al-houiti, Bang-Yen Chen, and Siraj Uddin. "Existence and uniqueness theorems for pointwise slant immersions in complex space forms." Filomat 35, no. 9 (2021): 3127–38. http://dx.doi.org/10.2298/fil2109127a.

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An isometric immersion f : Mn ? ?Mm from an n-dimensional Riemannian manifold Mn into an almost Hermitian manifold ?Mm of complex dimension m is called pointwise slant if its Wirtinger angles define a function defined on Mn. In this paper we establish the Existence and Uniqueness Theorems for pointwise slant immersions of Riemannian manifolds Mn into a complex space form ?Mn(c) of constant holomorphic sectional curvature c, which extend the Existence and Uniqueness Theorems for slant immersions proved by B.-Y. Chen and L. Vrancken in 1997.

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4

Alarcón, Antonio, та Franc Forstnerič. "Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve". Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, № 740 (1 липня 2018): 77–109. http://dx.doi.org/10.1515/crelle-2015-0069.

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Abstract We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{C}^{3}} (i.e. {u_{1}} has vanishing flux). If furthermore u is nonflat, then {u_{1}} can be chosen to have any prescribed flux and to be complete.

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5

Chen, Jingyi, and Ailana Fraser. "Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface." Canadian Journal of Mathematics 62, no. 6 (December 14, 2010): 1264–75. http://dx.doi.org/10.4153/cjm-2010-068-1.

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AbstractLet L be an oriented Lagrangian submanifold in an n-dimensional Kähler manifold M. Let u: D → M be a minimal immersion from a disk D with u(𝜕D) ⊂ L such that u(D) meets L orthogonally along u(𝜕D). Then the real dimension of the space of admissible holomorphic variations is at least n + μ(E, F), where μ(E, F) is a boundary Maslov index; the minimal disk is holomorphic if there exist n admissible holomorphic variations that are linearly independent over ℝ at some point p ∈ 𝜕D; if M = ℂPn and u intersects L positively, then u is holomorphic if it is stable, and its Morse index is at least n + μ(E, F) if u is unstable.

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6

EKHOLM, TOBIAS, JOHN ETNYRE, and MICHAEL SULLIVAN. "ORIENTATIONS IN LEGENDRIAN CONTACT HOMOLOGY AND EXACT LAGRANGIAN IMMERSIONS." International Journal of Mathematics 16, no. 05 (May 2005): 453–532. http://dx.doi.org/10.1142/s0129167x05002941.

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We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n + 1)-space from ℤ2 to ℤ. We demonstrate how the ℤ-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into ℂn and again including orientations strengthens the results. More precisely, we prove that the number of double points of an exact Lagrangian immersion of a closed manifold M whose associated Legendrian embedding has good DGA is at least half of the dimension of the homology of M with coefficients in an arbitrary field if M is spin and in ℤ2 otherwise.

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7

CHEN, B. Y., F. DILLEN, L. VERSTRAELEN, and L. VRANCKEN. "Lagrangian isometric immersions of a real-space-form Mn(c) into a complex-space-form M˜n(4c)." Mathematical Proceedings of the Cambridge Philosophical Society 124, no. 1 (July 1998): 107–25. http://dx.doi.org/10.1017/s030500419700217x.

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It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

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8

Marbes, Heinz. "On the Set of Umbilical Points of a Twistor-Holomorphic Immersion IntoR4." Mathematische Nachrichten 142, no. 1 (1989): 19–26. http://dx.doi.org/10.1002/mana.19891420103.

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9

Hasegawa, Kazuyuki. "The first Chern class and conformal area for a twistor holomorphic immersion." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 84, no. 1 (January 22, 2014): 67–83. http://dx.doi.org/10.1007/s12188-014-0089-3.

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10

Ando, Naoya. "Local characterizations of complex curves in C2 and sphere Schwarz maps." International Journal of Mathematics 27, no. 08 (July 2016): 1650067. http://dx.doi.org/10.1142/s0129167x16500671.

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In this paper, we obtain a local characterization of a complex curve in [Formula: see text] in terms of the induced metric and a holomorphic cubic differential [Formula: see text], and we see that any complex curve in [Formula: see text] is locally represented as the image by the composition of an affine Schwarz map and a parallel translation in [Formula: see text]. We obtain a local characterization of a sphere Schwarz immersion in terms of a positive-valued function [Formula: see text].

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11

GRUNDLAND, A. M., та İ. YURDUŞEN. "SURFACES OBTAINED FROM ℂPN-1 SIGMA MODELS". International Journal of Modern Physics A 23, № 32 (30 грудня 2008): 5137–57. http://dx.doi.org/10.1142/s0217751x08042699.

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In this paper, the Weierstrass technique for harmonic maps S2 → ℂPN-1 is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the ℂPN-1 model equations are defined on the sphere S2 and the associated action functional of this model is finite, then the generalized Weierstrass formula for immersion describes conformally parametrized surfaces in the su (N) algebra. In particular, for any holomorphic or antiholomorphic solution of this model the associated surface can be expressed in terms of an orthogonal projector of rank (N - 1). The implementation of this method is presented for two-dimensional conformally parametrized surfaces immersed in the su (3) algebra. The usefulness of the proposed approach is illustrated with examples, including the dilation-invariant meron-type solutions and the Veronese solutions for the ℂP2 model. Depending on the location of the critical points (zeros and poles) of the first fundamental form associated with the meron solution, it is shown that the associated surfaces are semiinfinite cylinders. It is also demonstrated that surfaces related to holomorphic and mixed Veronese solutions are immersed in ℝ8 and ℝ3, respectively.

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12

RODGERS, V. G. J. "QCD INSTANTONS AND 2D SURFACES." Modern Physics Letters A 07, no. 11 (April 10, 1992): 1001–8. http://dx.doi.org/10.1142/s0217732392000896.

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Some time ago, Atiyah showed that there exists a natural identification between the k-instantons of a Yang-Mills theory with gauge group G and the holomorphic maps from CP1 to ΩG. Since then, Nair and Mazur have associated the Θ vacua structure in QCD with self-intersecting Riemann surfaces immersed in four dimensions. From here they concluded that these 2D surfaces correspond to the non-perturbative phase of QCD and carry the topological information of the Θ vacua. In this paper we would like to elaborate on this point by making use of Atiyah’s identification. We will argue that an effective description of QCD may be more like a WZW model coupled to the induced metric of an immersion of a 2D Riemann surface in R4. We make some further comments on the relationship between the coadjoint orbits of the Kac-Moody group on G and instantons with axial symmetry and monopole charge.

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13

Némethi, András, and Gergő Pintér. "Immersions associated with holomorphic germs." Commentarii Mathematici Helvetici 90, no. 3 (2015): 513–41. http://dx.doi.org/10.4171/cmh/363.

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14

Verbitsky, Misha. "Wirtinger numbers and holomorphic symplectic immersions." Selecta Mathematica 10, no. 4 (April 2005): 551–59. http://dx.doi.org/10.1007/s00029-004-0268-7.

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15

Mok, Ngaiming. "On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature." Science in China Series A: Mathematics 48, S1 (December 2005): 123–45. http://dx.doi.org/10.1007/bf02884700.

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16

Cao, Huai-Dong, and Ngaiming Mok. "Holomorphic immersions between compact hyperbolic space forms." Inventiones Mathematicae 100, no. 1 (December 1990): 49–61. http://dx.doi.org/10.1007/bf01231180.

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17

Furuhata, Hitoshi, and Hiroshi Matsuzoe. "Holomorphic centroaffine immersions and the Lelieuvre correspondence." Results in Mathematics 33, no. 3-4 (May 1998): 294–305. http://dx.doi.org/10.1007/bf03322089.

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18

ALARCÓN, ANTONIO, and FRANC FORSTNERIČ. "NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY." Journal of the Australian Mathematical Society 106, no. 03 (August 23, 2018): 287–341. http://dx.doi.org/10.1017/s1446788718000125.

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Анотація:

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$ , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

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19

Kolarič, Dejan. "Parametric H-principle for holomorphic immersions with approximation." Differential Geometry and its Applications 29, no. 3 (June 2011): 292–98. http://dx.doi.org/10.1016/j.difgeo.2011.04.028.

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20

Ali, Akram, and Ali Alkhaldi. "Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications." Symmetry 11, no. 2 (February 11, 2019): 200. http://dx.doi.org/10.3390/sym11020200.

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Анотація:

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.

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21

Labourie, François. "Immersions isométriques elliptiques et courbes pseudo-holomorphes." Journal of Differential Geometry 30, no. 2 (1989): 395–424. http://dx.doi.org/10.4310/jdg/1214443596.

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22

Dor, Avner. "Immersions and embeddings in domains of holomorphy." Transactions of the American Mathematical Society 347, no. 8 (August 1, 1995): 2813–49. http://dx.doi.org/10.1090/s0002-9947-1995-1282885-5.

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23

Forstnerič, Franc. "Proper holomorphic immersions into Stein manifolds with the density property." Journal d'Analyse Mathématique 139, no. 2 (October 2019): 585–96. http://dx.doi.org/10.1007/s11854-019-0068-9.

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24

Castro-Infantes, Ildefonso, and Brett Chenoweth. "Carleman approximation by conformal minimal immersions and directed holomorphic curves." Journal of Mathematical Analysis and Applications 484, no. 2 (April 2020): 123756. http://dx.doi.org/10.1016/j.jmaa.2019.123756.

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25

Fiedler, Thomas. "Twistor holomorphic immersions of real surfaces into K�hler surfaces." Mathematische Annalen 282, no. 2 (June 1988): 337–42. http://dx.doi.org/10.1007/bf01456979.

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26

Bolton, J., L. M. Woodward та L. Vrancken. "Minimal immersions of S2 and ℝP2 into ℂPn with few higher order singularities". Mathematical Proceedings of the Cambridge Philosophical Society 111, № 1 (січень 1992): 93–101. http://dx.doi.org/10.1017/s0305004100075186.

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Анотація:

In this paper we extend ideas developed in 2, 4 to study certain minimal immersions of S2 and ℝP2 into ℂPn. Here S2 denotes the unit sphere in ℝ3 with its standard conformal structure and ℝP2 is S2 factored out by the antipodal map, while ℂPn denotes complex projective n-space equipped with the FubiniStudy metric of constant holomorphic sectional curvature 4. Since ℝPn with its standard metric of constant curvature 1 is included in ℂPn as a totally geodesic submanifold, this includes the case of minimal immersions into the unit sphere Sn(1) with its standard metric.

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27

Fang, Fuquan, and Sérgio Mendonça. "Complex immersions in Kähler manifolds of positive holomorphic $k$-Ricci curvature." Transactions of the American Mathematical Society 357, no. 9 (March 25, 2005): 3725–38. http://dx.doi.org/10.1090/s0002-9947-05-03675-5.

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28

Rigoli, Marco. "A Rigidity Result for Holomorphic Immersions of Surfaces in C P n." Proceedings of the American Mathematical Society 93, no. 2 (February 1985): 317. http://dx.doi.org/10.2307/2044769.

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29

Biswas, Indranil, and A. K. Raina. "Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space." Journal of Geometry and Physics 58, no. 10 (October 2008): 1417–28. http://dx.doi.org/10.1016/j.geomphys.2008.06.001.

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30

To, Wing-Keung. "Total geodesy of proper holomorphic immersions between complex hyperbolic space forms of finite volume." Mathematische Annalen 297, no. 1 (September 1993): 59–84. http://dx.doi.org/10.1007/bf01459488.

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31

Robaszewska, Maria. "A local characterization of affine holomorphic immersions with an anti-complex and ∇-parallel shape operator." Annales Polonici Mathematici 78, no. 1 (2002): 59–84. http://dx.doi.org/10.4064/ap78-1-7.

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32

Larusson, Finnur, and Tyson Ritter. "Proper holomorphic immersions in homotopy classes of maps from finitely connected planar domains into CxC*." Indiana University Mathematics Journal 63, no. 2 (2014): 367–83. http://dx.doi.org/10.1512/iumj.2014.63.5206.

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33

Rigoli, Marco. "A rigidity result for holomorphic immersions of surfaces in ${\bf C}{\rm P}\sp n$." Proceedings of the American Mathematical Society 93, no. 2 (February 1, 1985): 317. http://dx.doi.org/10.1090/s0002-9939-1985-0770544-3.

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34

Abe, Naoto, and Sanae Kurosu. "A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions." Results in Mathematics 44, no. 1-2 (August 2003): 3–24. http://dx.doi.org/10.1007/bf03322907.

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35

Hao, Yihong, An Wang, and Liyou Zhang. "On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space." Journal of Mathematical Analysis and Applications 423, no. 1 (March 2015): 547–60. http://dx.doi.org/10.1016/j.jmaa.2014.09.080.

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36

Leschke, K., and K. Moriya. "The $$\mu $$-Darboux transformation of minimal surfaces." manuscripta mathematica 162, no. 3-4 (September 12, 2019): 537–58. http://dx.doi.org/10.1007/s00229-019-01142-9.

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Abstract The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated $$\mathbb { C}_*$$C∗-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called $$\mu $$μ-Darboux transforms. We show that a $$\mu $$μ-Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a $$\mu $$μ-Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in $$\mathbb { C}\mathbb { P}^3$$CP3 which is canonically associated to a minimal surface $$f_{p,q}$$fp,q in the right-associated family of f. Here we use an extension of the notion of the associated family $$f_{p,q}$$fp,q of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at $$\mu =1$$μ=1. Moreover, the family of Willmore surfaces $$\mu $$μ-Darboux transforms, $$\mu \in \mathbb { C}_*$$μ∈C∗, extends to a $$\mathbb { C}\mathbb { P}^1$$CP1 family of Willmore surfaces $$f^\mu : M \rightarrow S^4$$fμ:M→S4 where $$\mu \in \mathbb { C}\mathbb { P}^1$$μ∈CP1.

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37

Bismut, J. M., and X. Ma. "Holomorphic immersions and equivariant torsion forms." Journal für die reine und angewandte Mathematik (Crelles Journal) 2004, no. 575 (January 4, 2004). http://dx.doi.org/10.1515/crll.2004.079.

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38

Foo, Wei Guo, and Joël Merker. "Holomorphic Immersions of Bi-Disks into 9D Real Hypersurfaces with Levi Signature (2,2)." International Mathematics Research Notices, October 1, 2020. http://dx.doi.org/10.1093/imrn/rnaa266.

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Анотація:

Abstract Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan’s method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$D real-analytic real hypersurface $M^{9}\subset \mathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.

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39

Bonsante, Francesco, та Christian El Emam. "On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences". International Mathematics Research Notices, 21 липня 2021. http://dx.doi.org/10.1093/imrn/rnab189.

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Abstract We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including $SL(2,\mathbb{C})$ and the space of geodesics of $\mathbb{H}^3$, and we prove a Gauss–Codazzi theorem in this setting. This approach has some interesting geometric consequences: (1) it provides a model for the transitioning of hypersurfaces among $\mathbb{H}^n$, ${\mathbb{A}}\textrm{d}{\mathbb{S}}^n$, $\textrm{d}{\mathbb{S}}^n$, and ${\mathbb{S}}^n$; (2) it provides an effective tool to construct holomorphic maps to the $\textrm{SO}(n,\mathbb{C})$-character variety, bringing to a simpler proof of the holomorphicity of the complex landslide; and (3) it leads to a correspondence, under certain hypotheses, between complex metrics on a surface (i.e., complex bilinear forms of its complexified tangent bundle) and pairs of projective structures with the same holonomy. Through Bers theorem, we prove a uniformization theorem for complex metrics.

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40

Bhowmick, Aritra. "On horizontal immersions of discs in fat distributions of type (4,6)." Journal of Topology and Analysis, October 9, 2021, 1–29. http://dx.doi.org/10.1142/s1793525321500539.

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In this paper, we discuss horizontal immersions of discs in certain corank-2 fat distributions on 6-dimensional manifolds. The underlying real distribution of a holomorphic contact distribution on a complex 3 manifold belongs to this class. The main result presented here says that the associated nonlinear PDE is locally invertible. Using this we prove the existence of germs of embedded horizontal discs.

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41

Hou, Zelin, and Enchao Bi. "Remarks on regular quantization and holomorphic isometric immersions on Hartogs triangles." Archiv der Mathematik, March 5, 2022. http://dx.doi.org/10.1007/s00013-022-01718-0.

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