In this note we show that on any compact subdomain of a K?hler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer ...In this note we show that on any compact subdomain of a K?hler 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 K?hler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be 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 earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.展开更多
基金partially supported by ERC Consolidator Grant IPFLOW(Grant No.725967)supported by the Academy of Finland(Finnish Centre of Excellence in Inverse Problems Research(Grant Nos.284715 and 309963))+2 种基金by the European Research Council under FP7/2007-2013 ERC StG(Grant No.307023)Horizon 2020 ERC CoG(Grant No.770924)partially supported by Australian Research Council(Grant Nos.DP190103302 and DP190103451)
文摘In this note we show that on any compact subdomain of a K?hler 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 K?hler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be 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 earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.
