Collapsing of the line bundle mean curvature flow on Kähler surfaces

Abstract

We study the line bundle mean curvature flow on Kähler surfaces under the hypercritical phase and a certain semipositivity condition. We naturally encounter such a condition when considering the blowup of Kähler surfaces. We show that the flow converges smoothly to a singular solution to the deformed Hermitian-Yang-Mills equation away from a finite number of curves of negative self-intersection on the surface. As an application, we obtain a lower bound of a Kempf-Ness type functional on the space of potential functions satisfying the hypercritical phase condition.

arXiv comment19 pages, final version, to appear in Calc. Var. Partial Differential Equations

References in corpus (6)

• (1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions
• Moment maps, nonlinear PDE, and stability in mirror symmetry
• The deformed Hermitian Yang-Mills equation on three-folds
• The J-flow on Kahler surfaces: a boundary case
• Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature
• The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics

Cited by in corpus (4)

• Stability and the deformed Hermitian-Yang-Mills equation
• Deformed Hermitian-Yang-Mills Equation on Compact Hermitian Manifolds
• Chern number inequalities of deformed Hermitian-Yang-Mills metrics on four dimensional Kaehler manifolds
• Examples of dHYM connections in a variable background