Paper · Oct 26, 2025

Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

Arunima Bhattacharya, Ravi Shankar, Jeremy Wall, Diego Yepez

As on arXiv

Abstract

We establish interior estimates for singularities of the Lagrangian mean curvature flow when the Lagrangian phase is critical, i.e., $∣Θ∣ \geq (n - 2) \frac{π}{2}$ , and extend our results to the broader class of Lagrangian mean curvature type equations. Our gradient estimates require certain structural conditions, and we construct $C^{α}$ singular viscosity solutions to show that criticality of the phase is necessary, and that these conditions cannot be removed in dimension one. We also introduce a new method for proving $C^{2, α}$ estimates by exponentiating the arctangent operator into a concave one when $∣Θ∣ \geq (n - 2) \frac{π}{2}$ and $n > 2$ .

Mean curvature flow Lagrangian mean curvature flow Singularities Weak & singular flows

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Lagrangian mean curvature flow Wider flows In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume… Start with: A canonical way to deform a Lagrangian submanifold (Smoczyk, 1996) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase

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