Tohoku Mathematical Journal

Weighted Hamiltonian stationary Lagrangian submanifolds and generalized Lagrangian mean curvature flows in toric almost Calabi-Yau manifolds

Hikaru Yamamoto

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In this paper, we generalize examples of Lagrangian mean curvature flows constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. To be more precise, we construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized Lagrangian mean curvature flows starting from these examples. We allow these flows to have some singularities and topological changes.

Article information

Tohoku Math. J. (2), Volume 68, Number 3 (2016), 329-347.

Received: 22 April 2014
Revised: 24 October 2014
First available in Project Euclid: 23 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Lagrangian mean curvature flow special Lagrangian submanifold


Yamamoto, Hikaru. Weighted Hamiltonian stationary Lagrangian submanifolds and generalized Lagrangian mean curvature flows in toric almost Calabi-Yau manifolds. Tohoku Math. J. (2) 68 (2016), no. 3, 329--347. doi:10.2748/tmj/1474652263.

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  • T. Behrndt, Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein, Complex and Differential Geometry, Springer Proceedings in Mathematics, 8, 65–79, Springer-Verlag, 2011.
  • T. Behrndt, Mean curvature flow of Lagrangian submanifolds with isolated conical singularities, arXiv:1107.4803v1, 2011.
  • K. A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, Princeton University Press, 1978.
  • V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Mathematics, 122, Birkhäuser Boston, Inc., Boston, MA, 1994.
  • R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157.
  • D. Joyce, Special Lagrangian $m$-folds in $\mathbb{C}^m$ with symmetries, Duke Math. J. 115 (2002), no. 1, 1–51.
  • D. Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62.
  • Y. I. Lee and M.-T. Wang, Hamiltonian stationary cones and self-similar solutions in higher dimensions, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1491–1503.
  • A. Mironov, On new examples of Hamiltonian-minimal and minimal Lagrangian submanifolds in $\mathbb{C}^m$ and $\mathbb{CP}^m$, Sb. Math. 195 (2004), no. 1, 85–96.
  • A. Mironov and T. Panov, Hamiltonian-minimal Lagrangian submanifolds in toric varieties, Russian Math. Surveys 68 (2013), no. 2, 392–394.
  • A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), nos. 1–2, 243–259.
  • R. P. Thomas and S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113.
  • H. Yamamoto, Special Lagrangians and Lagrangian self-similar solutions in cones over toric Sasaki manifolds, New York J. Math. 22 (2016), 501–526.