2025/05/27 by Wentao Cao, Jonas Hirsch, Dominik Inauen · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.1007/s00526-025-03019-0
openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
Abstract For any θ lt;(1)/(3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> , we show that very weak solutions to the two-dimensional Monge–Ampère equation with regularity C1,θ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> </mml:math> are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.