0000000001039469

AUTHOR

Colin Guillarmou

showing 3 related works from this author

The linearized Calderón problem on complex manifolds

2019

International audience; 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.…

Class (set theory)Pure mathematicsGeneral MathematicsHolomorphic function01 natural sciencesinversio-ongelmatSet (abstract data type)symbols.namesake[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]0101 mathematics[MATH]Mathematics [math]complex manifoldMathematics::Symplectic GeometryMathematicsosittaisdifferentiaaliyhtälötCalderón problemMathematics::Complex VariablesApplied MathematicsRiemann surface010102 general mathematicsLimitingStandard methodsManifold010101 applied mathematicsHarmonic function[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]symbolsinverse problemMathematics::Differential Geometrymonistot
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The X-Ray Transform for Connections in Negative Curvature

2016

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e. vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connect…

Mathematics - Differential GeometryPure mathematicsHermitian bundlesGeodesic[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]Connection (vector bundle)Boundary (topology)Dynamical Systems (math.DS)X-ray transforms01 natural sciencesinversio-ongelmatHiggs fieldsTensor fieldMathematics - Analysis of PDEsFOS: MathematicsSectional curvatureMathematics - Dynamical Systems0101 mathematicsmath.APMathematical PhysicsPhysicsX-ray transformParallel transport010102 general mathematicsStatistical and Nonlinear Physicsconnections010101 applied mathematicsHiggs fieldmath.DGDifferential Geometry (math.DG)[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]Mathematics::Differential Geometrymath.DSAnalysis of PDEs (math.AP)[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]Communications in Mathematical Physics
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The linearized Calder\'on problem on complex manifolds

2018

In this note we show that on any compact subdomain of a K\"ahler 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 Calder\'on problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of K\"ahler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot by treated by standard methods for the Calder\'on problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends resu…

Mathematics - Differential GeometryMathematics - Analysis of PDEsMathematics::Complex VariablesMathematics - Complex VariablesMathematics::Differential GeometryMathematics::Symplectic Geometry
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