2024/04/10 by José Edson Sampaio, Sampaio, José Edson
Engineering · Mathematics · #14B05 #32S50 #Advanced Banach Space Theory #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2404.06943
openalex publication_date 2024/04/10 · openalex created_date 2024/04/12 · openalex updated_date 2026/07/28
In this paper, we prove metric analogues, in any dimension and in any co-dimension, of the famous Theorem of Mumford on smoothness of normal surfaces and the beautiful Theorem of Ramanujam that gives a topological characterization of ℂ2 as an algebraic surface. For instance, we prove that a complex analytic set that is log-Lipschitz regular at 0 (i.e., a complex analytic set that has a neighbourhood of the origin which bi-log-Lipschitz homeomorphic to an Euclidean ball) must be smooth at 0. We prove even more, we prove that if a complex analytic set X such that, for each 0<α<1, (X,0) and (ℝk,0) are bi-α-Hölder homeomorphic, then X must be smooth at 0. These results generalize the Lipschitz Regularity Theorem, which says that a Lipschitz regular complex analytic set must be smooth. Global versions of these results are also presented here and, in particular, we obtain a characterization of an affine linear subspace as a pure-dimensional entire complex analytic set.