The Doctrine and Covenants, Section 45:59
59 For the Lord shall be in their midst, and his glory shall be upon them, and he will be their king and their lawgiver.
The Doctrine and Covenants, Section 84:19-22
19 And this greater priesthood administereth the gospel and holdeth the key of the mysteries of the kingdom, even the key of the knowledge of God.
20 Therefore, in the ordinances thereof, the power of godliness is manifest.
21 And without the ordinances thereof, and the authority of the priesthood, the power of godliness is not manifest unto men in the flesh;
22 For without this no man can see the face of God, even the Father, and live.
History of the Church, Joseph Smith 5:212
Christ and the resurrected Saints will reign over the earth during the thousand years. They will not probably dwell upon the earth, but will visit it when they please or when it is necessary to govern it. There will be wicked men on the earth during the thousand years. The heathen nations who will not come up to worship will be visited with the judgments of God, and must eventually be destroyed from the earth.
Journal of Discourses, Brigham Young 1:31
The Millennium consists in this–every heart in the Church and Kingdom of God being united in one; the Kingdom increasing to the overcoming of everything opposed to the economy of heaven, and Satan being bound, and having a seal set upon him. All things else will be as they are now, we shall eat, drink, and wear clothing.
Journal of Discourses, John Taylor 11:14
We look forward to the time, and try to help it on, when God will assert his own right with regard to the government of the earth; when, as in religious matters so in political matters, he will enlighten the minds of those that bear rule, he will teach the kings wisdom and instruct the senators by the Spirit of eternal truth; when to him “every knee shall bow and every tongue confess that Jesus is the Christ.” Then “shall the earth be full of knowledge, like as the waters cover the sea.” Then shall the mists of darkness be swept away by the light of eternal truth. Then will the intelligence of Heaven beam forth on the human mind, and by it they will comprehend everything that is great, and good, and glorious.
Journal of Discourses, John Taylor 6:3
What is the kingdom of God? It is God’s government upon the earth and in heaven. What is his Priesthood? It is the rule, authority, administration, if you please, of the government of God on the earth or in the heavens; for the same Priesthood that exists upon the earth exists in the heavens, and that Priesthood holds the keys of the mysteries of the revelations of God; and the legitimate head of that Priesthood, who has communion with God, is the Prophet, Seer, and Revelator to his Church and people on the earth. When the will of God is done on earth as it is in heaven, that Priesthood will be the only legitimate ruling power under the whole heavens; for every other power and influence will be subject to it. When the millennium which we have been speaking of is introduced, all potentates, powers, and authorities–every man, woman, and child will be in subjection to the kingdom of God; they will be under the power and dominion of the Priesthood of God: then the will of God will be done on the earth as it is done in heaven.
Scriptural Teachings of the Prophet Joseph Smith :268-269
Christ and the resurrected Saints will reign over the earth during the thousand years. They will not probably dwell upon the earth, but will visit it when they please or when it is necessary to govern it.