Parallel mean curvature surfaces with constant contact angle along free boundaries

We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension reduction theorem for branched immersed bordered Riemann surfaces of higher genus with multiple boundary components under the same parallel mean curvature and constant contact angle assumptions. Furthermore, we construct a family of explicit examples of branched minimal immersions satisfying the non-orthonormal constant contact angle free boundary condition, which demonstrate the sharpness of both the classification result and the codimension reduction result.