The structure of CSC-1 reducible conformal metrics on a compact Riemann surface with finite conical singularities

We study the structure of CSC-1 reducible conformal metrics on a compact Riemann surface with finite conical singularities. We prove that any compact Riemann surface with a CSC-1 reducible conformal metric of finite conical singularities can be divided into a finite number of pieces by cutting along geodesics where each piece is isometric to some football. This allows us to study the existence of CSC-1 reducible conformal metrics with finite conical singularities on a compact Riemann surface. As an application, we give an angle condition of the existence of CSC-1 reducible conformal metrics with finite conical singularities on a compact Riemann surface.Comment: 43pages, 13page