Éomer fears any enemies remaining near its walls might find a way into Minas Tirith and do harm there, and has asked you to patrol its perimeter.
'There are many ways to sneak into the City unnoticed. After we set forth for the Black Gate, too few eyes will be keeping watch and even fewer swords and bows will be defending it. Please patrol the outer walls of Minas Tirith and vouchsafe no remaining enemies loiter there.'
Head to the north section of the outer wall of after-battle Minas Tirith.
Patrol the outer wall of after-battle Minas Tirith.
Patrol outside the main gate of after-battle Minas Tirith.
Head south to continue patrolling the outer wall of after-battle Minas Tirith.
Head to the southern point of after-battle Minas Tirith's outer wall.
You should talk to Éomer, who can be found at Aragorn's pavilion on the after-battle Pelennor.