2018/05/02 by Colin Guillarmou, Mikko Salo, Guillarmou, Colin +3
Mathematics · #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory #math.AP #math.CV #math.DG
paper · pdf · doi:10.48550/arxiv.1805.00752
15 pages
arxiv created 2018/05/02 · openalex publication_date 2018/05/02 · arxiv updated 2018/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot by treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends results of Guillarmou and Tzou (Duke Math. J. 2011) from the case of Riemann surfaces to higher dimensional complex manifolds.