What did Sauron think the free peoples did with his ring following the events on Amon Hen?

Question

Originally Sauron probably assumed the free peoples wanted to let the One Ring remain in Rivendell, as he didn't think they would try to bring it to Mordor to destroy it. But on Amon Hen (near the Argonath) Frodo put on the ring to escape Boromir, and Sauron saw him on Amon Hen. What did Sauron think from there on? Didn't he get suspicious or did he think the ring actually draws the Fellowship to Sauron himself? Or did he think they would bring it to Rohan or to Minas Tirith to be safe?

And why didn't Sauron send the Nazgûl on their fellbeasts to Amon Hen after learning the One Ring is there?

Answer 1

I think the movie interpreted this scene differently from how it was meant to be interpreted. Here's the passage from "The Breaking of the Fellowship", the final chapter of The Fellowship of the Ring:

And suddenly he felt the Eye. There was an eye in the Dark Tower that did not sleep. He knew that it had become aware of his gaze. A fierce eager will was there. It leaped towards him; almost like a finger he felt it, searching for him. Very soon it would nail him down, know just exactly where he was. Amon Lhaw it touched. It glanced upon Tol Brandir--he threw himself from the seat, crouching, covering his head with his grey hood.

The only thing for certain seems to be that Sauron knew Frodo was wearing the Ring, and in which direction he lay. There's no indication that Sauron knew, as the shadow approached that he was getting closer. I interpret this to mean only when Sauron's gaze would fall on Amon Hen would he see Frodo. (As an analogy: imagine looking for an ant that you know is somewhere on a sidewalk. As your gaze moves down the walk, you know the ant is there somewhere, but you won't know for sure where until you actually see it. You won't know how far the ant is from any given spot on the ground simply by looking at it.)

He heard himself crying out: Never, never! Or was it: Verily I come, I come to you? He could not tell. Then as a flash from some other point of power there came to his mind another thought: Take it off! Fool, take it off! Take off the Ring!

Listen to Gandalf, Frodo!*

The two powers strove in him. For a moment, perfectly balanced between their piercing points, he writhed, tormented. Suddenly he was aware of himself again. Frodo, neither the Voice nor the Eye: free to choose, and with one remaining instant in which to do so. He took the Ring off his finger. He was kneeling in clear sunlight before the high seat. A black shadow seemed to pass like an arm above him; it missed Amon Hen and groped out west, and faded. Then all the sky was clean and blue and birds sang in every tree.

Sauron's gaze did not reach Amon Hen until after Frodo had removed the Ring, and so never "saw" him. He may have immediately known that Frodo had removed the Ring, but for all he knew, he had not yet looked far enough, rather than having just passed over head.

Note that the Orcs that killed Boromir and captured Merry and Pippen were already near by, and probably not sent there because Sauron was or had become aware of the company's presence. It is telling that Grishnákh, later in "The Uruk-hai" (chapter 3 of The Two Towers) mentions a winged Nazgûl in the vicinity, yet it was not dispatched to Amon Hen.

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* (From Chapter 5, "The White Rider", in The Two Towers, when Gandalf is speaking to Aragorn, Gimli, and Legolas:

"... Very nearly [the Ring] was revealed to the Enemy, but it escaped. I had some part in that: for I sat in a high place, and I strove with the Dark Tower; and the Shadow passed. ..."

I thought perhaps the "high place" was Zirak-Zigil, where Gandalf fell defeating the Balrog and where he was returned after a spell. According to Appendix B, though, Gandalf arrived in Lóthlorien on February 17, and Frodo's adventure on Amon Hen was on February 26. The "high place" is probably not to be taken as a mountain top, though perhaps a tall hill in Lóthlorien is a reasonable assumption.

)

Answer 2

In addition to the excellent answers already provided, remember that from Sauron's perspective after the incident on Amon Hen, he knows the following events took place:

Edited to add item #0:

0. Saruman captured two hobbits on the banks of the Anduin and refused to send them to Mordor, but chose instead to keep them for himself.

1. Saruman radically stepped up his war on Rohan.
2. Saruman stopped communicating with him using the palantir.
3. Isengard was destroyed.
4. A hobbit suddenly appeared in the palantir.
5. Aragorn contested the palantir with him.
6. Gandalf showed up at Minas Tirith - as the White and not the Grey.
7. The Witch King somehow was defeated at the Field of Pelennor.
8. An army marched north from Minas Tirith, loudly proclaiming that the King of Gondor had returned.

From his perspective, using the imperfect information at his disposal, the only conclusion he can draw is that at one time a hobbit possessed the Ring - but that Saruman, Aragorn and possibly Gandalf took the Ring from that hobbit and then fought over it, with Aragorn winning in the end. Saruman was destroyed in that conflict, with Gandalf becoming the new Ring-Lord's servant or lieutenant.

Answer 3

Since we are rarely privy to Sauron's actual thoughts or point of view, if we are to trust Gandalf as having special insight into Sauron's frame of mind, I don't think we are given that Sauron knows "exactly" where Frodo was when he, Frodo, had the ring on at that time.

Frodo does "hear" himself crying out:

"Never! Never! Or was it: Verily, I come, I come to you? He could not tell.
_{Fellowship of the Ring, Chapter 10, The Breaking of the Fellowship}

Later, Gandalf says:

Sauron...knows that this precious thing which he lost has been found again, but he does not yet know where it is, or so we hope. And therefore, he is now in great doubt. For if we have found this thing, there are some among us with the strength to wield it. That too he knows.
_{The Return of the King: Chapter 9, The Last Debate}