JOEL - The Book of Mormon tells of a man called Hagoth who lead a sea-faring group of people out of the west sea in a course northward and were never heard of again, nor was it known where they went. In fact the rest of the Nephites assumed that they might have been "drowned in the depths of the sea. "(Alma 63:5-8)
No one knows for sure if the Polynesians are descendents of the Book of Mormon people, although some church leaders have strongly expressed their opinions on this. According to Bishop Robert L. Simpson speaking to BYU students:
"I would like to tell you young people that in a great gathering of Polynesians held right in Salt Lake City just prior to 1915, a prophet of the Lord, President Joseph F. Smith, addressing a group of Polynesians who had come to Salt Lake City to participate in their temple endowment work, made the statement that without a doubt this man Hagoth and his company were the progenitors of the Polynesian races, and that this migration was the beginning of the Polynesian population in the South Pacific." (Bishop Robert L. Simpson, April 4, 1962)
In speaking to the Hawaiians at Laie, Elder Matthew Cowley said: "Brothers and sisters, you are God's children—-you are Israel. You have in your veins the blood of Nephi" (Cole, William A., and Edwin W. Jensen. Israel in the Pacific: A Genealogical Text For Polynesia. Salt Lake City: Genealogical Society of Utah, 1961).
Elder Mark E. Petersen, in his conference message of 1962, said: "The Polynesian Saints are characterized by a tremendous faith. Why do they have this great faith? It is because these people are of the blood of Israel. They are heirs to the promises of the Book of Mormon. God is now awakening them to their great destiny. As Latter-day Saints we have always believed that the Polynesians are descendants of Lehi and blood relatives of the American Indians, despite the contrary theories of other men (Petersen, Mark E. "New Evidence for the Book of Mormon," Improvement Era (June 1962) 65:456-59; also in Conference Report (Apr 1962) 111-15.).
Despite these opinions we must however consider them in light of a statement from the First Presidency. In a letter to the mission president of the Samoan Mission dated September 6, 1972, and signed by N. Eldon Tanner and Marion G. Romney, under the letterhead of the First Presidency, they wrote: "In your letter of September 6, 1972, you ask if the Polynesian people are Lamanites or Nephites. There has been much speculation about the origin of these people. We have, however, no scriptural evidence or revelation from the Lord that would tell us exactly where these people came from or their background."
Many Book of Mormon scholars have also considered the possibility that some of these people made it to the Polynesian islands, based partly on research by Thor Heyerdahl; some Polynesian oral traditions; and a lot of conjecture(See Sidney B. Sperry, Book of Mormon Compendium, [Deseret Book Co., 1972], p. 369.)
So far, there has been no solid scientific or archaeological evidence found to support a Nephite heritage for the Polynesians.
But if they are descendants of the Nephites, it is logical to assume that they might have kept some kind of record of their history, but it may have been lost or just not yet found.
In 1976, President Spencer W. Kimball said: "I thought to read to you a sacred scripture which pertains especially to you the islanders of the Pacific. It is in the sixty-third chapter of Alma. . . . [He then read the account of Hagoth.]And so it seems to me rather clear that your ancestors moved northward and crossed a part of the South Pacific. You did not bring your records with you, but you brought much food and provisions." (Kimball, Spencer W. "Official Report of the Samoa Area Conference Held in Pago Pago and Apia, Samoa, February 15, 16, 17, 18, 1976)
So acording to President Kimball we don't know for sure if they even had any records. The bottom line is that we just don't know if there are any ancient Polynesian records to be found. Perhaps at the second coming of Christ their history will be revealed and we will know what really happened.