Minimizing area among Lagrangian surfaces: The mapping problem

By R. Schoen, J. Wolfson. 2000

166 citations recorded by the source corpus.

This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the langrangian surfaces of a Kaehler surface, or, more generally, a symplectic 4-manifold with suitable metric, and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of the homology generated by lagrangian cycles).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/967b27d3567d32f23bbb935196716c3117e6db8e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. W. Minicozzi
2. T. Colding
3. D. Joyce
4. W. H. Meeks
5. Jiayu Li
6. Jingyi Chen
7. Knut Smoczyk
8. Joaquín Pérez

In Geometry and Topology

1. W. Minicozzi
2. T. Colding
3. D. Joyce
4. W. H. Meeks
5. Jiayu Li
6. Jingyi Chen
7. Knut Smoczyk
8. Reto Muller

Citations in the corpus

• Self-similar solutions and translating solitons for Lagrangian mean curvature flow (2008, unknown): “A related method for constructing special Lagrangians by minimizing volume amongst Lagrangians using Geometric Measure Theory was proposed by Schoen and Wolfson [23].”
• Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow (2014, background): “Wolfson [72] constructed an example of a Calabi–Yau 2-fold (M,J, g,Ω) (a K3 surface) with the following properties: (i) There exists α ∈ H2(M,Z) with α · α = −4, such that every compact, immersed Lagrangian L in M has [L] ∈ Z · α ⊂ H2(M,Z).”
• Width and finite extinction time of Ricci flow (2007, methodology): “We will now use the conformality of the map ht to control the energy of the composition as in Proposition 5:4 of [34].”
• Singularity of mean curvature flow of Lagrangian submanifolds (2003, methodology): “fold is called almost calibrated by ReΩ if cosθ>0. Constructing minimal Lagrangian submanifolds is an important but very challenging task. In a compact Ka¨hler-Einstein surface, Schoen and Wolfson [ScW] have shown the existence of a branched surface which minimizes area among Lagrangian competito”