What is the php function for “user’s public profile”

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For all author meta details you can use the_author_meta

http://codex.wordpress.org/Template_Tags/the_author_meta

<a href="https://wordpress.stackexchange.com/questions/37317/<?php the_author_meta("user_url', $current_user->ID);?>"><?php the_author_meta('display_name', $current_user->ID);?></a>

If its for user meta use get_user_meta

http://codex.wordpress.org/Function_Reference/get_user_meta

<a href="https://wordpress.stackexchange.com/questions/37317/<?php echo get_user_meta($current_user->ID,"user_url');?>"><?php echo get_user_meta($current_user->ID, 'display_name');?></a>