When Hakadosh Baruch Hu first created the world, everything was "tohu va'vohu" (Breishis 1:2). By the end of the sixth day of creation, most of the tohu va'vohu was replaced with a beautifully developed world. Some, however, was still left behind for man to join with Hashem in completing His creation (see Breishis 2:3). Amalek is part of the remaining tohu va'vohu[1], and we were instructed that we should not assume that all of the evil in the world will go away by itself; but rather we have an obligation to fight against evil, and complete Hashem's briah by removing all of the remaining tohu va'vohu.
The chumash tells us (Shemos 17:16) that Hashem swore by His throne that we would have an everlasting and ongoing campaign against Amalek. Both Hashem's name and His throne are mentioned there in abbreviated form (see Rashi there) to indicate that until Amalek will ultimately be obliterated, His name and throne are incomplete. When we observe contradictions between the moral, ethical, and religious G-d given principles and the "real" world (for example - we see righteous people who suffer and wicked people who prosper), it seems to indicate that Hashem's throne is "incomplete". It appears to us as if (kevayachol) He is not in full control of the world and is unable to see to it that His wish should prevail.
The full name of Hashem (Havaya) indicates that in the past, present, and future He is the all-powerful One. As long as "Amalek" is still around, it appears as if He was only all-powerful in the past, and will be so in the future, but is not all-powerful in the present. In the shortened version of Hashem's name (Koh) the yud represents the future, the heh represents the past, and the missing heh and vuv (which should have been in the middle) represents the fact that it appears as if (kevayachol) He were not all powerful in the present[2].
It is because of this mistaken perception that the name of the city of Yerushalayim is almost always spelled (in tanach as well as in kesubos and gittin) missing a yud. Yerushalayim is the official capital of Eretz Yisroel[3], and Hashem is officially the King of all of Eretz Yisroel. That is why Yerushalayim is referred to as "the throne of Hashem" (Yirmiyahu 3:17). Because the throne today seems to be incomplete, therefore Yerushalayim is always spelled in an incomplete form[4].
After all the evil forces in the world (Amalek) will be overcome and there will no longer be any contradiction between the natural world and G-d's moral, ethical, and religious principles, then the power of His kingship will be apparent, His throne will be complete, and the name of His capital city (Yerushalayim) will be spelled in its complete form. We will refer to Hashem by His full name (Havaya), as opposed to His abbreviated name (Koh), and the prophecy of Zechariah (14:9) that Hashem will only have one name (Havaya) will be fulfilled. Not only will we believe, as we always have, that the One and only G-d is omnipotent, but that will be our perception as well[5].
When we recite Shema we say "baruch shem kevod malchuso" in a whisper. We don't proclaim Hashem's kingship out loud because we don't readily perceive it. On Yom Kippur, when we think about tikkun of the world, the custom is to recite "baruch shem" out loud. The day will yet come when Hashem's Malchus will no longer be whispered about, but will be clear and obvious[6].
[1] See note #108 in Ish Hahalacha
[2] See Mepeninei Horav, Parshas Vaera
[3] See Rambam, Hilchos Melachim 1:5
[4] See Tzlach to Berachos 58a and Bikvei Natzonp. 224
[5] See Pesachim 50a
[6] See Mepeninei Harav, Tefilos Rosh Hashana v'Yom Kippur, section 5