Alexandrov-type theorem for singular capillary CMC hypersurfaces in the half-space

In this paper, we consider the classification problem for critical points of relative isoperimetric-type problem in the half-space. Under certain regularity assumption, we prove an Alexandrov-type theorem for the singular capillary CMC hypersurfaces in the half-space. The key ingredient is a new shifted distance function that is suitable for the study of capillary problem in the half-space.