2025/12/08 by Cairo, Hannah, Zhang, Ruixiang
Mathematics · #42B37 #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.2512.08064
openalex publication_date 2025/12/08 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28
We find a family of compact Ck hypersurfaces where the local Mizohata-Takeuchi Conjecture fails with a power loss of Rα for any α<(n-1)/(n-1+k). Moreover, this family is dense in the Ck topology, and so the local Mizohata-Takeuchi conjecture fails for many convex hypersurfaces. In particular, the local Mizohata-Takeuchi Conjecture fails with a power loss of Rα for any α<(n-1)/(n+1) for many C2 convex hypersurfaces. This power matches the best known upper bound in a paper by Tony Carbery, Marina Iliopoulou and Hong Wang up to the endpoint. For the proof, our weight is positive definite as in the first author's recent log(R)-loss counterexample, and our construction is based on a projection of a higher rank lattice. As a by-product, we also construct compact convex C2 hypersurfaces whose rescaling contains many lattice points in any dimension.