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Longtime existence of the Lagrangian mean curvature flow


Given a compact Lagrangian submanifold in flat space evolving by its mean curvature, we prove uniform \(C^{2,\alpha}\)-bounds in space and C 2-estimates in time for the underlying Monge-Ampére equation under weak and natural assumptions on the initial Lagrangian submanifold. This implies longtime existence and convergence of the Lagrangian mean curvature flow. In the 2-dimensional case we can relax our assumptions and obtain two independent proofs for the same result.

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Correspondence to Knut Smoczyk.

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Received: 3 September 2002, Accepted: 12 June 2003, Published online: 4 September 2003

Mathematics Subject Classification (2000):


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Smoczyk, K. Longtime existence of the Lagrangian mean curvature flow. Cal Var 20, 25–46 (2004).

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  • Curvature Flow
  • Lagrangian Submanifold
  • Natural Assumption
  • Flat Space
  • Independent Proof