Display Custom User Profile Fields in a Block on All of Their Published Pages
You should take a look at the Context module.


  Context allows you to manage contextual conditions and reactions for
  different portions of your site. You can think of each context as
  representing a "section" of your site. For each context, you can
  choose the conditions that trigger this context to be active and
  choose different aspects of Drupal that should react to this active
  context.
  
  Think of conditions as a set of rules that are checked during page
  load to see what context is active. Any reactions that are associated
  with active contexts are then fired.


Or a simpler solution might be to change your path from "their-name/content" to "user/their-name/content" and only show the block on "user/*/content".