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Journal articles on the topic 'Minimal Lagrangian Submanifolds'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Iriyeh, Hiroshi. "Symplectic topology of Lagrangian submanifolds of ℂPn with intermediate minimal Maslov numbers." Advances in Geometry 17, no. 2 (March 28, 2017): 247–64. http://dx.doi.org/10.1515/advgeom-2017-0005.

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AbstractWe examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in ℂPn. First we give cohomological constraints on a Lagrangian submanifold in ℂPn whose first integral homology is p-torsion. In the case where (n, p) = (5,3), (8, 3), we prove that the cohomologies with coefficients in ℤ2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)ℤ3) and SU(3)/ℤ3, respectively. Then we calculate the Floer cohomology with coefficients in ℤ2 of a monotone Lagrangian submanifold SU(p)/ℤp in ${\mathbb C}P^{p^2-1}.$

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2

Bolton, J., C. Rodriguez Montealegre, and L. Vrancken. "Characterizing warped-product Lagrangian immersions in complex projective space." Proceedings of the Edinburgh Mathematical Society 52, no. 2 (May 28, 2009): 273–86. http://dx.doi.org/10.1017/s0013091507000922.

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AbstractStarting from two Lagrangian immersions and a horizontal curve in S3(1), it is possible to construct a new Lagrangian immersion, which we call a warped-product Lagrangian immersion. In this paper, we find two characterizations of warped-product Lagrangian immersions. We also investigate Lagrangian submanifolds which attain at every point equality in the improved version of Chen's inequality for Lagrangian submanifolds of ℂPn(4) as discovered by Opreaffi We show that, for n≥4, an n-dimensional Lagrangian submanifold in ℂPn(4) for which equality is attained at all points is necessarily minimal.

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3

Blair, David E. "On Lagrangian Catenoids." Canadian Mathematical Bulletin 50, no. 3 (September 1, 2007): 321–33. http://dx.doi.org/10.4153/cmb-2007-031-4.

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AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

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4

Ohnita, Yoshihiro. "Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces." Complex Manifolds 6, no. 1 (January 1, 2019): 303–19. http://dx.doi.org/10.1515/coma-2019-0016.

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AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

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5

Tevdoradze, Z. "The Hörmander and Maslov Classes and Fomenko's Conjecture." gmj 4, no. 2 (April 1997): 185–200. http://dx.doi.org/10.1515/gmj.1997.185.

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Abstract Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.

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6

CHEN, BANG-YEN. "Ideal Lagrangian immersions in complex space forms." Mathematical Proceedings of the Cambridge Philosophical Society 128, no. 3 (May 2000): 511–33. http://dx.doi.org/10.1017/s0305004199004247.

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Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. In this paper we study Lagrangian immersions in complex space forms which are ideal. We prove that all Lagrangian ideal immersions in a complex space form are minimal. We also determine ideal Lagrangian submanifolds in complex space forms.

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7

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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8

Butscher, Adrian. "Deformations of minimal Lagrangian submanifolds with boundary." Proceedings of the American Mathematical Society 131, no. 6 (October 24, 2002): 1953–64. http://dx.doi.org/10.1090/s0002-9939-02-06800-4.

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9

Mironov, Andrei E., and Taras E. Panov. "Hamiltonian-minimal Lagrangian submanifolds in toric varieties." Russian Mathematical Surveys 68, no. 2 (April 30, 2013): 392–94. http://dx.doi.org/10.1070/rm2013v068n02abeh004835.

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10

Maccheroni, Roberta. "Complex analytic properties of minimal Lagrangian submanifolds." Journal of Symplectic Geometry 18, no. 4 (2020): 1127–46. http://dx.doi.org/10.4310/jsg.2020.v18.n4.a6.

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11

Anciaux, Henri. "Minimal Lagrangian submanifolds in indefinite complex space." Illinois Journal of Mathematics 56, no. 4 (2012): 1331–43. http://dx.doi.org/10.1215/ijm/1399395835.

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12

Kajigaya, Toru. "Reductions of minimal Lagrangian submanifolds with symmetries." Mathematische Zeitschrift 289, no. 3-4 (November 29, 2017): 1169–89. http://dx.doi.org/10.1007/s00209-017-1992-y.

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13

Li, Haizhong, Hui Ma, Joeri Van der Veken, Luc Vrancken, and Xianfeng Wang. "Minimal Lagrangian submanifolds of the complex hyperquadric." Science China Mathematics 63, no. 8 (November 25, 2019): 1441–62. http://dx.doi.org/10.1007/s11425-019-9551-2.

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14

Miyaoka, Reiko, and Yoshihiro Ohnita. "Lagrangian geometry of the Gauss images of isoparametric hypersurfaces in spheres." Complex Manifolds 6, no. 1 (January 1, 2019): 265–78. http://dx.doi.org/10.1515/coma-2019-0013.

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AbstractThe Gauss images of isoparametric hypersufaces of the standard sphere Sn+1 provide a rich class of compact minimal Lagrangian submanifolds embedded in the complex hyperquadric Qn(ℂ). This is a survey article based on our joint work [17] to study the Hamiltonian non-displaceability and related properties of such Lagrangian submanifolds.

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15

Chen, B. Y., F. Dillen, L. Verstraelen, and L. Vrancken. "An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 1 (1996): 153–65. http://dx.doi.org/10.1017/s0308210500030651.

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In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.

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16

Lotay, Jason D., and Tommaso Pacini. "From minimal Lagrangian to J-minimal submanifolds: persistence and uniqueness." Bollettino dell'Unione Matematica Italiana 12, no. 1-2 (October 31, 2018): 63–82. http://dx.doi.org/10.1007/s40574-018-0183-z.

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17

Lotay, Jason D., and Felix Schulze. "Consequences of Strong Stability of Minimal Submanifolds." International Mathematics Research Notices 2020, no. 8 (May 9, 2018): 2352–60. http://dx.doi.org/10.1093/imrn/rny095.

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Abstract In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang [12] directly extends to the enhanced Brakke flows of Ilmanen [5]. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow, which are proved to occur by Neves [7].

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18

Draper, Chris, and Ian McIntosh. "Minimal Lagrangian submanifolds via the geodesic Gauss map." Communications in Analysis and Geometry 24, no. 5 (2016): 969–91. http://dx.doi.org/10.4310/cag.2016.v24.n5.a3.

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19

Castro, Ildefonso, Haizhong Li, and Francisco Urbano. "Hamiltonian-minimal Lagrangian submanifolds in complex space forms." Pacific Journal of Mathematics 227, no. 1 (September 1, 2006): 43–63. http://dx.doi.org/10.2140/pjm.2006.227.43.

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20

Castro, Ildefonso, Cristina R. Montealegre, and Francisco Urbano. "Minimal Lagrangian submanifolds in the complex hyperbolic space." Illinois Journal of Mathematics 46, no. 3 (July 2002): 695–721. http://dx.doi.org/10.1215/ijm/1258130980.

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21

Opozda, Barbara. "A moduli space of minimal affine Lagrangian submanifolds." Annals of Global Analysis and Geometry 41, no. 4 (October 16, 2011): 535–47. http://dx.doi.org/10.1007/s10455-011-9298-5.

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22

Palmer, Bennett. "Biharmonic capacity and the stability of minimal Lagrangian submanifolds." Tohoku Mathematical Journal 55, no. 4 (December 2003): 529–41. http://dx.doi.org/10.2748/tmj/1113247128.

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23

Rybnikov, I. P. "Minimal Lagrangian submanifolds in ℂP n with diagonal metric." Siberian Mathematical Journal 52, no. 1 (January 2011): 105–12. http://dx.doi.org/10.1134/s0037446606010113.

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24

Dong, Yuxin. "Hamiltonian-minimal Lagrangian submanifolds in Kaehler manifolds with symmetries." Nonlinear Analysis: Theory, Methods & Applications 67, no. 3 (August 2007): 865–82. http://dx.doi.org/10.1016/j.na.2006.06.045.

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25

Li, Haizhong, Hui Ma, and Guoxin Wei. "A class of minimal Lagrangian submanifolds in complex hyperquadrics." Geometriae Dedicata 158, no. 1 (June 2, 2011): 137–48. http://dx.doi.org/10.1007/s10711-011-9625-9.

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26

Bolton, J., and L. Vrancken. "Ruled Minimal Lagrangian Submanifolds of Complex Projective 3-Space." Asian Journal of Mathematics 9, no. 1 (2005): 45–56. http://dx.doi.org/10.4310/ajm.2005.v9.n1.a4.

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27

Alqahtani, Lamia Saeed. "Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms." Advances in Mathematical Physics 2021 (September 20, 2021): 1–7. http://dx.doi.org/10.1155/2021/6260639.

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In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.

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28

Urban, Zbyněk, and Ján Brajerčík. "The fundamental Lepage form in variational theory for submanifolds." International Journal of Geometric Methods in Modern Physics 15, no. 06 (May 8, 2018): 1850103. http://dx.doi.org/10.1142/s0219887818501037.

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The multiple-integral variational functionals for finite-dimensional immersed submanifolds are studied by means of the fundamental Lepage equivalent of a homogeneous Lagrangian, which can be regarded as a generalization of the well-known Hilbert form in the classical mechanics. The notion of a Lepage form is extended to manifolds of regular velocities and plays a basic role in formulation of the variational theory for submanifolds. The theory is illustrated on the minimal submanifolds problem, including analysis of conservation law equations.

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29

DONG, Yuxin. "On second order minimal Lagrangian submanifolds in complex space forms." Science in China Series A 48, no. 11 (2005): 1505. http://dx.doi.org/10.1360/04ys0197.

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30

Anciaux, Henri, and Ildefonso Castro. "Construction of Hamiltonian-Minimal Lagrangian submanifolds in Complex Euclidean Space." Results in Mathematics 60, no. 1-4 (June 2, 2011): 325–49. http://dx.doi.org/10.1007/s00025-011-0148-3.

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31

Amarzaya, Amartuvshin, and Yoshihiro Ohnita. "Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces." Tohoku Mathematical Journal 55, no. 4 (December 2003): 583–610. http://dx.doi.org/10.2748/tmj/1113247132.

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32

Castro, Ildefonso, and Luc Vrancken. "Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation." Results in Mathematics 40, no. 1-4 (October 2001): 130–43. http://dx.doi.org/10.1007/bf03322703.

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33

Ali, Akram, Pişcoran Laurian-Ioan, Ali H. Alkhaldi, and Lamia Saeed Alqahtani. "Ricci curvature on warped product submanifolds of complex space forms and its applications." International Journal of Geometric Methods in Modern Physics 16, no. 09 (September 2019): 1950142. http://dx.doi.org/10.1142/s0219887819501421.

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The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

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34

Vrancken, Luc. "Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms." Proceedings of the American Mathematical Society 130, no. 5 (October 17, 2001): 1459–66. http://dx.doi.org/10.1090/s0002-9939-01-06213-x.

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35

Dong, Yuxin, and Yingbo Han. "Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms." Nonlinear Analysis: Theory, Methods & Applications 66, no. 5 (March 2007): 1091–99. http://dx.doi.org/10.1016/j.na.2006.01.007.

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36

Oh, Yong-Geun. "Second variation and stabilities of minimal lagrangian submanifolds in K�hler manifolds." Inventiones Mathematicae 101, no. 1 (December 1990): 501–19. http://dx.doi.org/10.1007/bf01231513.

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37

Goldstein, Edward. "A Construction of New Families of Minimal Lagrangian Submanifolds via Torus Actions." Journal of Differential Geometry 58, no. 2 (June 2001): 233–61. http://dx.doi.org/10.4310/jdg/1090348326.

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38

Kriele, Marcus, and Luc Vrancken. "Minimal Lagrangian submanifolds of Lorentzian complex space forms with constant sectional curvature." Archiv der Mathematik 72, no. 3 (March 1999): 223–32. http://dx.doi.org/10.1007/s000130050326.

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39

LI, XingXiao. "On the correspondence between symmetric equiaffine hyperspheres and the minimal symmetric Lagrangian submanifolds." SCIENTIA SINICA Mathematica 44, no. 1 (January 1, 2014): 13–36. http://dx.doi.org/10.1360/012013-155.

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40

Hijazi, O., S. Montiel, and F. Urbano. "Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds." Mathematische Zeitschrift 253, no. 4 (March 28, 2006): 821–53. http://dx.doi.org/10.1007/s00209-006-0936-8.

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41

FISH, JOEL W. "ESTIMATES FOR J-CURVES AS SUBMANIFOLDS." International Journal of Mathematics 22, no. 10 (October 2011): 1375–431. http://dx.doi.org/10.1142/s0129167x11007306.

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In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c2f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).

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42

ONO, Hajime. "Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian." Journal of the Mathematical Society of Japan 55, no. 1 (January 2003): 243–54. http://dx.doi.org/10.2969/jmsj/1196890852.

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43

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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44

Vérine, Alexandre. "Bohr–Sommerfeld Lagrangian submanifolds as minima of convex functions." Journal of Symplectic Geometry 18, no. 1 (2020): 333–53. http://dx.doi.org/10.4310/jsg.2020.v18.n1.a9.

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45

Bolton, J., C. Scharlach, and L. Vrancken. "From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space." Bulletin of the Australian Mathematical Society 66, no. 3 (December 2002): 465–75. http://dx.doi.org/10.1017/s0004972700040302.

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In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

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46

Castro, Ildefonso, and Francisco Urbano. "On a minimal Lagrangian submanifold of C n foliated by spheres." Michigan Mathematical Journal 46, no. 1 (May 1999): 71–82. http://dx.doi.org/10.1307/mmj/1030132359.

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47

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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48

Cerne, Miran. "Minimal discs with free boundaries in a Lagrangian submanifold of C^n." Indiana University Mathematics Journal 44, no. 1 (1995): 0. http://dx.doi.org/10.1512/iumj.1995.44.1982.

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49

Pérez, Joaquín, and Antonio Ros. "The space of complete minimal surfaces with finite total curvature as lagrangian submanifold." Transactions of the American Mathematical Society 351, no. 10 (February 8, 1999): 3935–52. http://dx.doi.org/10.1090/s0002-9947-99-02250-3.

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50

Bedulli, Lucio, and Anna Gori. "A Hamiltonian Stable Minimal Lagrangian Submanifold of Projective Space with Nonparallel Second Fundamental Form." Transformation Groups 12, no. 4 (November 27, 2007): 611–17. http://dx.doi.org/10.1007/s00031-007-0060-9.

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