2024/11/10 by Mathew George, George, Mathew
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2411.06497
A complex Monge-Ampère equation for differential (p,p)-forms is introduced on compact Kähler manifolds. For any 1 ≤ p < n, we show the existence of smooth solutions unique up to adding constants. For p=1, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for p=n-1, this gives the Monge-Ampère equation for (n-1) plurisubharmonic functions studied by Tosatti-Weinkove. For other p values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to (p,p)-forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.