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Journal articles on the topic 'Lagrangian surfaces'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Zhao, Yaomin, Yue Yang, and Shiyi Chen. "Evolution of material surfaces in the temporal transition in channel flow." Journal of Fluid Mechanics 793 (March 23, 2016): 840–76. http://dx.doi.org/10.1017/jfm.2016.152.

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We report a Lagrangian study on the evolution of material surfaces in the Klebanoff-type temporal transitional channel flow. Based on the Eulerian velocity field from the direct numerical simulation, a backward-particle-tracking method is applied to solve the transport equation of the Lagrangian scalar field, and then the isosurfaces of the Lagrangian field can be extracted as material surfaces in the evolution. Three critical issues for Lagrangian investigations on the evolution of coherent structures using material surfaces are addressed. First, the initial scalar field is uniquely determined based on the proposed criteria, so that the initial material surfaces can be approximated as vortex surfaces, and remain invariant in the initial laminar state. Second, the evolution of typical material surfaces initially from different wall distances is presented, and then the influential material surface with the maximum deformation is identified. Large vorticity variations with the maximum curvature growth of vortex lines are also observed on this surface. Moreover, crucial events in the transition can be characterized in a Lagrangian approach by conditional statistics on the material surfaces. Finally, the influential material surface, which is initially a vortex surface, is demonstrated as a surrogate of the vortex surface before significant topological changes of vortical structures. Therefore, this material surface can be used to elucidate the continuous temporal evolution of vortical structures in transitional wall-bounded flows in a Lagrangian perspective. The evolution of the influential material surface is divided into three stages: the formation of a triangular bulge from an initially disturbed streamwise–spanwise sheet, rolling up of the vortex sheet near the bulge ridges with the vorticity intensification and the generation and evolution of signature hairpin-like structures with self-induced dynamics of vortex filaments.

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2

PAVLOTSKY, I. P., and M. STRIANESE. "SOME PECULIAR PROPERTIES OF THE DARWIN’S LAGRANGIAN." International Journal of Modern Physics B 09, no. 23 (October 20, 1995): 3069–83. http://dx.doi.org/10.1142/s0217979295001166.

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In the post-Galilean approximation the Lagrangians are singular on a submanifold of the phase space. It is a local singularity, which differs from the ones considered by Dirac. The dynamical properties are essentially peculiar on the studied singular surfaces. In the preceding publications,1,2,3 two models of singular relativistic Lagrangians and the rectilinear motion of two electrons, determined by Darwin’s Lagrangian, were examined. In the present paper we study the peculiar dynamical properties of the two-dimensional Darwin’s Lagrangian. In particular, it is shown that the minimal distance between two electrons (the so called “radius of electron”) appears in the two-dimensional motion as well as in one-dimensional case. Some new peculiar properties are discovered.

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3

Carriazo, Alfonso, Verónica Martín-Molina, and Luc Vrancken. "Null pseudo-isotropic Lagrangian surfaces." Colloquium Mathematicum 150, no. 1 (2017): 87–101. http://dx.doi.org/10.4064/cm7107s-12-2016.

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4

Kossowski, Marek. "Prescribing invariants of Lagrangian surfaces." Topology 31, no. 2 (April 1992): 337–47. http://dx.doi.org/10.1016/0040-9383(92)90026-e.

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5

Kawasaki, Morimichi. "Superheavy Lagrangian immersions in surfaces." Journal of Symplectic Geometry 17, no. 1 (2019): 239–49. http://dx.doi.org/10.4310/jsg.2019.v17.n1.a5.

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6

Hind, Richard. "Lagrangian unknottedness in Stein surfaces." Asian Journal of Mathematics 16, no. 1 (2012): 1–36. http://dx.doi.org/10.4310/ajm.2012.v16.n1.a1.

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7

YANG, YUE, and D. I. PULLIN. "On Lagrangian and vortex-surface fields for flows with Taylor–Green and Kida–Pelz initial conditions." Journal of Fluid Mechanics 661 (October 1, 2010): 446–81. http://dx.doi.org/10.1017/s0022112010003125.

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For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions.

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8

Bektaş, Burcu, Marilena Moruz, Joeri Van der Veken, and Luc Vrancken. "Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 from minimal surfaces in 𝕊3." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 03 (December 27, 2018): 655–89. http://dx.doi.org/10.1017/prm.2018.43.

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AbstractWe study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

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9

Craizer, Marcos. "Equiaffine characterization of Lagrangian surfaces in ℝ4." International Journal of Mathematics 26, no. 09 (August 2015): 1550074. http://dx.doi.org/10.1142/s0129167x15500743.

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For non-degenerate surfaces in ℝ4, a distinguished transversal bundle called affine normal plane bundle was proposed in [K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in ℝ4, Internat. J. Math. 4(1) (1993) 127–165]. Lagrangian surfaces have remarkable properties with respect to this normal bundle, like for example, the normal bundle being Lagrangian. In this paper, we characterize those surfaces which are Lagrangian with respect to some parallel symplectic form in ℝ4.

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10

HASHIMOTO, YOSHITAKE, and KIYOSHI OHBA. "CUTTING AND PASTING OF RIEMANN SURFACES WITH ABELIAN DIFFERENTIALS I." International Journal of Mathematics 10, no. 05 (August 1999): 587–617. http://dx.doi.org/10.1142/s0129167x99000239.

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We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ωΛ of the second kind on the Riemann surface. Using Λ and ωΛ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

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11

SATO, Noriaki. "On Lagrangian surfaces in CP2(c̃)." Hokkaido Mathematical Journal 31, no. 2 (February 2002): 441–51. http://dx.doi.org/10.14492/hokmj/1350911873.

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12

Evans, J. D. "Lagrangian spheres in Del Pezzo surfaces." Journal of Topology 3, no. 1 (2010): 181–227. http://dx.doi.org/10.1112/jtopol/jtq004.

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13

Etgü, Tolga, David McKinnon, and B. Doug Park. "Lagrangian tori in homotopy elliptic surfaces." Transactions of the American Mathematical Society 357, no. 9 (March 31, 2005): 3757–74. http://dx.doi.org/10.1090/s0002-9947-05-03757-8.

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14

Mettler, Thomas. "Minimal Lagrangian connections on compact surfaces." Advances in Mathematics 354 (October 2019): 106747. http://dx.doi.org/10.1016/j.aim.2019.106747.

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15

Han, Qing, and Guofang Wang. "Hessian surfaces and local Lagrangian embeddings." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 35, no. 3 (May 2018): 675–85. http://dx.doi.org/10.1016/j.anihpc.2017.07.003.

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16

Ma, Hui. "Hamiltonian Stationary Lagrangian Surfaces in ℂP2." Annals of Global Analysis and Geometry 27, no. 1 (March 2005): 1–16. http://dx.doi.org/10.1007/s10455-005-5214-1.

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17

Opozda, Barbara. "Flat affine Lagrangian surfaces in C2." Differential Geometry and its Applications 27, no. 3 (June 2009): 430–41. http://dx.doi.org/10.1016/j.difgeo.2009.01.004.

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18

Dorfmeister, Josef F., Walter Freyn, Shimpei Kobayashi, and Erxiao Wang. "Survey on real forms of the complex A2(2)-Toda equation and surface theory." Complex Manifolds 6, no. 1 (January 1, 2019): 194–227. http://dx.doi.org/10.1515/coma-2019-0011.

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AbstractThe classical result of describing harmonic maps from surfaces into symmetric spaces of reductive Lie groups [9] states that the Maurer-Cartan form with an additional parameter, the so-called loop parameter, is integrable for all values of the loop parameter. As a matter of fact, the same result holds for k-symmetric spaces over reductive Lie groups, [8].In this survey we will show that to each of the five different types of real forms for a loop group of A2(2) there exists a surface class, for which some frame is integrable for all values of the loop parameter if and only if it belongs to one of the surface classes, that is, minimal Lagrangian surfaces in ℂℙ2, minimal Lagrangian surfaces in ℂℍ2, timelike minimal Lagrangian surfaces in ℂℍ12, proper definite affine spheres in ℝ3 and proper indefinite affine spheres in ℝ3, respectively.

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19

Vidussi, Stefano. "Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori." Journal of Differential Geometry 74, no. 3 (November 2006): 507–22. http://dx.doi.org/10.4310/jdg/1175266235.

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20

Chen, Bang-Yen. "CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE." Proceedings of the Edinburgh Mathematical Society 48, no. 2 (May 23, 2005): 337–64. http://dx.doi.org/10.1017/s0013091504000203.

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AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

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21

SPELIOTOPOULOS, ACHILLES D., and HARRY L. MORRISON. "ON THE KOSTERLITZ–THOULESS TRANSITION ON COMPACT RIEMANN SURFACES." Modern Physics Letters B 07, no. 03 (February 10, 1993): 171–82. http://dx.doi.org/10.1142/s0217984993000199.

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A Lagrangian for the two-dimensional vortex gas is derived from a general microscopic Lagrangian for 4 He atoms on an arbitrary compact Riemann Surface without boundary. In the constant density limit the vortex Hamiltonian obtained from this Lagrangian is found to be the same as the Kosterlitz and Thouless Coulombic interaction Hamiltonian. The partition function for the Kosterlitz–Thouless ensemble on the general compact is formulated and mapped into the sine–Gordon field theory.

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22

CASTRO, ILDEFONSO, and FRANCISCO URBANO. "On twistor harmonic surfaces in the complex projective plane." Mathematical Proceedings of the Cambridge Philosophical Society 122, no. 1 (July 1997): 115–29. http://dx.doi.org/10.1017/s030500419600117x.

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We completely classify all the twistor harmonic (non-minimal) Lagrangian immersions of compact surfaces in the complex projective plane [Copf ]ℙ2, i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over [Copf ]ℙ2 are harmonic maps.

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23

Dai, Bo, Chung-I. Ho, and Tian-Jun Li. "Nonorientable Lagrangian surfaces in rational 4–manifolds." Algebraic & Geometric Topology 19, no. 6 (October 20, 2019): 2837–54. http://dx.doi.org/10.2140/agt.2019.19.2837.

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24

Lai, Kuan-Wen, Yu-Shen Lin, and Luca Schaffler. "Decomposition of Lagrangian classes on K3 surfaces." Mathematical Research Letters 28, no. 6 (2021): 1739–63. http://dx.doi.org/10.4310/mrl.2021.v28.n6.a5.

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25

Chantraine, Baptiste. "Some non-collarable slices of Lagrangian surfaces." Bulletin of the London Mathematical Society 44, no. 5 (April 3, 2012): 981–87. http://dx.doi.org/10.1112/blms/bds026.

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26

Etgü, Tolga. "Symplectic and Lagrangian Surfaces in 4-Manifolds." Rocky Mountain Journal of Mathematics 38, no. 6 (December 2008): 1975–89. http://dx.doi.org/10.1216/rmj-2008-38-6-1975.

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27

Wang, Jun, and Xiaowei Xu. "Lagrangian surfaces in the complex hyperquadric Q2." Journal of Geometry and Physics 97 (November 2015): 61–68. http://dx.doi.org/10.1016/j.geomphys.2015.07.009.

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28

Qiu, Weiyang. "Non-orientable Lagrangian Surfaces with Controlled Area." Mathematical Research Letters 8, no. 6 (2001): 693–701. http://dx.doi.org/10.4310/mrl.2001.v8.n6.a1.

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29

Hofherr, Florian, and Daniel Karrasch. "Lagrangian Transport through Surfaces in Compressible Flows." SIAM Journal on Applied Dynamical Systems 17, no. 1 (January 2018): 526–46. http://dx.doi.org/10.1137/17m1132938.

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30

Chen, Bang-Yen. "Lagrangian minimal surfaces in Lorentzian complex plane." Archiv der Mathematik 91, no. 4 (September 29, 2008): 366–71. http://dx.doi.org/10.1007/s00013-008-2733-6.

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31

Mishachev, K. N. "The classification of lagrangian bundles over surfaces." Differential Geometry and its Applications 6, no. 4 (December 1996): 301–20. http://dx.doi.org/10.1016/s0926-2245(96)00024-1.

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32

Georgiou, Nikos. "On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space." MATHEMATICA SCANDINAVICA 111, no. 2 (December 1, 2012): 187. http://dx.doi.org/10.7146/math.scand.a-15224.

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We study area-stationary surfaces in the space $\mathbf{L}(\mathbf{H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in $\mathbf{L}(\mathbf{H}^3)$ is an area-stationary surface. We then classify Lagrangian area-stationary surfaces $\Sigma$ in $\mathbf{L}(\mathbf{H}^3)$ and prove that the family of parallel surfaces in $\mathbf{H}^3$ orthogonal to the geodesics $\gamma\in \Sigma$ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.

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33

Lee, Yng-Ing. "The deformation of Lagrangian minimal surfaces in Kähler-Einstein surfaces." Journal of Differential Geometry 50, no. 2 (1998): 299–330. http://dx.doi.org/10.4310/jdg/1214461172.

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34

Anciaux, Henri, Brendan Guilfoyle, and Pascal Romon. "Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface." Journal of Geometry and Physics 61, no. 1 (January 2011): 237–47. http://dx.doi.org/10.1016/j.geomphys.2010.09.017.

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35

KOCHAN, DENIS. "QUANTIZATION OF NON-LAGRANGIAN SYSTEMS." International Journal of Modern Physics A 24, no. 28n29 (November 20, 2009): 5319–40. http://dx.doi.org/10.1142/s0217751x0904748x.

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A novel method for quantization of non-Lagrangian (open) systems is proposed. It is argued that the essential object, which provides both classical and quantum evolution, is a certain canonical two-form defined in extended velocity space. In this setting classical dynamics is recovered from the stringy-type variational principle, which employs umbilical surfaces instead of histories of the system. Quantization is then accomplished in accordance with the introduced variational principle. The path integral for the transition probability amplitude (propagator) is rearranged to a surface functional integral. In the standard case of closed (Lagrangian) systems the presented method reduces to the standard Feynman's approach. The inverse problem of the calculus of variation, the problem of quantization ambiguity and the quantum mechanics in the presence of friction are analyzed in detail.

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36

SASAHARA, TORU. "BIHARMONIC LAGRANGIAN SURFACES OF CONSTANT MEAN CURVATURE IN COMPLEX SPACE FORMS." Glasgow Mathematical Journal 49, no. 3 (September 2007): 497–507. http://dx.doi.org/10.1017/s0017089507003886.

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AbstractBiharmonic Lagrangian surfaces of constant mean curvature in complex space forms are classified. A further important point is that new examples of marginally trapped biharmonic Lagrangian surfaces in an indefinite complex Euclidean plane are obtained. This fact suggests that Chen and Ishikawa's classification of marginally trapped biharmonic surfaces [6] is not complete.

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37

Lee, Yng-Ing. "Lagrangian minimal surfaces in Kähler–Einstein surfaces of negative scalar curvature." Communications in Analysis and Geometry 2, no. 4 (1994): 579–92. http://dx.doi.org/10.4310/cag.1994.v2.n4.a4.

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38

MAETA, SHUN, and HAJIME URAKAWA. "BIHARMONIC LAGRANGIAN SUBMANIFOLDS IN KÄHLER MANIFOLDS." Glasgow Mathematical Journal 55, no. 2 (February 25, 2013): 465–80. http://dx.doi.org/10.1017/s0017089512000730.

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AbstractWe give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian H-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.

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39

DEGANI, A. T., J. D. A. WALKER, and F. T. SMITH. "Unsteady separation past moving surfaces." Journal of Fluid Mechanics 375 (November 25, 1998): 1–38. http://dx.doi.org/10.1017/s0022112098001839.

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Unsteady boundary-layer development over moving walls in the limit of infinite Reynolds number is investigated using both the Eulerian and Lagrangian formulations. To illustrate general trends, two model problems are considered, namely the translating and rotating circular cylinder and a vortex convected in a uniform flow above an infinite flat plate. To enhance computational speed and accuracy for the Lagrangian formulation, a remeshing algorithm is developed. The calculated results show that unsteady separation is delayed with increasing wall speed and is eventually suppressed when the speed of the separation singularity approaches that of the local mainstream velocity. This suppression is also described analytically. Only ‘upstream-slipping’ separation is found to occur in the model problems. The changes in the topological features of the flow just prior to the separation that occur with increasing wall speed are discussed.

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40

Sheridan, Nick, and Ivan Smith. "Rational equivalence and Lagrangian tori on K3 surfaces." Commentarii Mathematici Helvetici 95, no. 2 (June 16, 2020): 301–37. http://dx.doi.org/10.4171/cmh/489.

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41

Charette, François. "Gromov width and uniruling for orientable Lagrangian surfaces." Algebraic & Geometric Topology 15, no. 3 (June 19, 2015): 1439–51. http://dx.doi.org/10.2140/agt.2015.15.1439.

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42

DENG, Shangrong. "Lagrangian H-Umbilical Surfaces in Complex Lorentzian Plane." International Electronic Journal of Geometry 9, no. 2 (October 30, 2016): 87–93. http://dx.doi.org/10.36890/iejg.584604.

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43

Karrasch, Daniel. "Lagrangian Transport Through Surfaces in Volume-Preserving Flows." SIAM Journal on Applied Mathematics 76, no. 3 (January 2016): 1178–90. http://dx.doi.org/10.1137/15m1051348.

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44

Lee, Yng-Ing. "The limit of Lagrangian surfaces in $R^4$." Duke Mathematical Journal 71, no. 2 (August 1993): 629–31. http://dx.doi.org/10.1215/s0012-7094-93-07124-4.

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45

Iriyeh, Hiroshi, and Takashi Sakai. "Tight Lagrangian surfaces in S 2 × S 2." Geometriae Dedicata 145, no. 1 (June 27, 2009): 1–17. http://dx.doi.org/10.1007/s10711-009-9398-6.

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46

Chang, Shaoping. "On Hamiltonian stable minimal Lagrangian surfaces in CP2." Journal of Geometric Analysis 10, no. 2 (June 2000): 243–55. http://dx.doi.org/10.1007/bf02921823.

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47

Mese, Chikako. "The Bernstein problem for complete Lagrangian stationary surfaces." Proceedings of the American Mathematical Society 129, no. 2 (July 27, 2000): 573–80. http://dx.doi.org/10.1090/s0002-9939-00-05603-3.

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48

Opozda, Barbara. "Locally symmetric minimal affine Lagrangian surfaces in C2." Monatshefte für Mathematik 156, no. 4 (August 7, 2008): 357–70. http://dx.doi.org/10.1007/s00605-008-0023-9.

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49

Hélein, Frédéric, and Pascal Romon. "Hamiltonian stationary Lagrangian surfaces in $\mathbb{C}^2$." Communications in Analysis and Geometry 10, no. 1 (2002): 79–126. http://dx.doi.org/10.4310/cag.2002.v10.n1.a5.

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50

Van, Le Khong. "MINIMAL Φ-LAGRANGIAN SURFACES IN ALMOST HERMITIAN MANIFOLDS." Mathematics of the USSR-Sbornik 67, no. 2 (February 28, 1990): 379–91. http://dx.doi.org/10.1070/sm1990v067n02abeh001368.

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