As the hero Frodo faces an outer quest and an inner one. Frodo must destroy the ring and overcome the spiritual struggle with evil within himself. There is a physical challenge and a spiritual one. We see into this struggle by seeing into the eyes of Frodo.
above... The fire of the ring penetrates into the eyes of Frodo
At bag end Frodo’s eyes are filled with shadow on Gandalf’s return
Frodo, as the hero, has a depth that other members of the fellowship do not have or are striving to gain. His suffering is deeper. The elements of fire and shadow pursue the Hobbit. Indeed fire and shadow have absorbed themselves into Frodo. The hobbit will have two formidable enemies to overcome in his soul. This battle will be harder fought than any. The origin of this fire and shadow is the most powerful dark force in middle earth. Fire and shadow are the elements that compose the eye
The depth of Frodo is in his eyes. The eyes convey the inner struggle of the ringbearer. The tears of Frodo convey this inner duel. Frodo offers the ring to Galadriel but the tears of the hobbit suggest he cannot ever let it go. Giving the ring to another would not end the deeper struggle with the Eye going on within. The ring is but a totem or talisman of the deeper struggle.
Redemption can never come by giving up the ring. The deep struggle can only be relieved by tears
The tears of the hero are so redeeming they can even bring out the deepest qualities of a future king:
above: Frodo and Aragorn share tears. For Frodo they herald more courage and for Aragorn a deeper sense of resolution
Frodo does not only carry the wound of the morgul blade. There is a deeper wound. The fire of the eye and the shadow of Mordor have already found Frodo in Bag End. The wound of the Nazgul and Frodo’s meetings with the Eye only confirm the inner struggle.