Yes, skating in itself is an interesting sport on the physics part.
Speed skaters use long straight blades, hinged at the toes to propel the stroke better, in the 400 mtr sprint, speeds of up to 70 Km/h are achieved!
Short track skaters use very thin curved blades, fixed but with high pedestals so they can really lean into those tight curves.
Hockey skaters use short thick blades, I assume to them both the rocker and radius of hollow are very important due to the manoeuvrability needed while battling it out in a match
Figure skaters also use short thick blades, but theirs can be thinner than hockey skaters since there is less wear and tear because no fighting on the ice, and the diverse disciplines have different requirements re curves and edges.
How does the actual area of contact surface differs between those branches of skating? There must a minimum contact surface area, otherwise you'd sink into the ice?
Hydroplaning is the key of skating, but has this actually been studied in great detail, i.e. is there a 'bow wave' to climb? I guess it is all very minute and any advantage/disadvantages are only marginal. But inquiring minds want to know ...
Track cycling materials, technique and mechanics have been studied very closely and affluent programs that have the means have bettered their results considerably though applied science.
Figure skating blades aren't thinner than hockey blades