  We construct higher genus Riemann's minimal surfaces properly embedded in the
Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples
to two halves genus zero Riemann's minimal surfaces. In first we need to
perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux
vector along the catenoidal ends. Then we study the mapping property of the
Jacobi operator on the half Riemann example as a perturbation analysis of a
CMC-Delaunay half cylinder.
