The regular site header is not automatically shared to the mobile site, but you can specify a header just for your mobile site.

The reason the main site header isn't automatically shared is becuase in most cases it would be too large for a mobile site.  It is best to make the header for your mobile site simple.  Perhaps your company logo and a slogan.

You can make the mobile site header and footer here: YourDomain.com > Mobile Version Settings

Once in this area you will see an area to add your mobile header and footer. 

See screenshot

Notice the editor here is a little more limited.  You do not have the copy frame to clipboard option or paste.  If you wish to copy content from another page I recommend copying the source and pasting here in source view.

You can also opt to not show the site menu at all for the mobile site. (by checking the "Don't Show Site Menu" check box)

Also you can enable a separate site menu for your mobile site. (by checking the "Separate Site Menu
for Mobile Version")

(NOTE: if you wish to have a link to the full site, you can link some text to a URL like this:http://www.YourDomain.com/home?mobileVersion=off ) (this links to an alias of "home")