The linearized Calderón problem on complex manifolds

Colin Guillarmou (1) , Mikko Salo (2) , Leo Tzou (3)

1 UP11 - Université Paris-Sud - Paris 11
2 JYU - Department of Mathematics and Statistics [Jyväskylä Univ]
3 The University of Sydney

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 holo-morphic functions with approximately prescribed critical points. This extends results of [GT11 GT11] from the case of Riemann surfaces to higher dimensional complex manifolds.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Colin Guillarmou, Mikko Salo, Leo Tzou. The linearized Calderón problem on complex manifolds. Acta Mathematica Sinica English Series, 2019, 35 (6), pp.1043-1056. ⟨hal-01827890⟩