To make sure Uttaran asked Arjuna to repeat his ten names for he had heard them earlier. Arjuna then told them to him in words reminiscent of the tenth chapter of the Gita.
Hanta teham samachakshe dasha namani yaani me
Vairate srunu twam yaani purva sruthaani te
Ekaagramaanaso bhutva srunu sarva samaahitaha
Arjunaha Phaalguno Jishnuhu Kiriti shvethavahanaha
Bhibhasurvijayaha krishnaha savyasachi dhananjaya
O son of Virata I will tell you my ten names, which you had heard earlier. Hear then with a concentrated mind and with full attention: Arjuna, Phalguna Vishnu, Kiritin, Shvetavahana, Bhibhatsu, Vijaya, Krishna Savyasachi, and Dhananjaya.
Why are you called Vijaya, Kiritin, Savyasachi? I know the meanings of all these names. If you tell me the reason for these names I will believe in you fully.
Having conquered all countries and collected their wealth I lived in the midst of riches which is why the call me Dhananjaya.
When I go to fight with invincible kings, I never return without defeating them so they call me Vijaya.
When I fight in the battlefield the horses yoked to my chariot are white and golden hued hence they call me Shvetavahana
I was born I the Himalayas when the nakshatra Uttara Phalguni was in the ascendant so they call me Phalguna.
Since a diadem brilliant like the sun was placed on my head by Indra after my battle with the Danavas so they call me Kiritin.
I have never committed a hateful action on the field of battle so I am known as Bibhatsu among men and celestials.
Both of my hands are capable of drawing the Gandiva, hence they know me as Savyasachi among men and celestials
My complexion is rare on earth and I perform pure deeds so I among known as Arjuna
I am unapproachable, irrepressible, dreadful and the son of the chastiser of Paka; so I am known as Jishnu among men and celestials.
Krishna, my tenth name was given to me by my father out of affection for a black boy of great purity.
After he heard all this the son of Virata approached Partha reverentially and said
“I am known as Bhuminjaya as well as Uttaran. By my good fortune I have seen you, Partha. Welcome, Dhananjaya! You who have red eyes and mighty arms resembling the trunks of elephants.! You should pardon me for whatever I said out of ignorance. You have performed many wonderful and difficult feats; my fears are now removed and I bear a great love for you. Mounting this wonderful chariot with me as your charioteer tell me which division do you wish to attack?”
I am pleased with you, I shall disperse all your enemies for you O best among men. Be at ease, O mighty armed hero, behold me fighting with your enemies in battle, performing fearful feats.
The story of how Arjuna single-handedly dispersed the Kaurava heroes is history.
But there is much that we may learn from the submission of Uttara to Arjuna, so similar to the later submission of Arjuna himself to Krishna.