Is Genesis 18 regarded as a Theophany of the Trinity by any denomination? Abraham addresses the three visitors as LORD. Is there any group of Christians that understands this as a theophany of the Trinity?


  Genesis 18:1-2 NASB 1 Now the LORD appeared to him by the oaks of Mamre, while he was sitting at the tent door in the heat of the day. 2 When he lifted up his eyes and looked, behold, three men were standing opposite him; and when he saw them, he ran from the tent door to meet them and bowed himself to the earth,
According to Augustine, it seems, he interpreted the three men in Genesis 18 as The Trinity God appearing in human form, hence the three men.

On the Trinity (Book II) - Chapter 10


  But under the oak at Mamre he saw three men, whom he invited, and
  hospitably received, and ministered to them as they feasted. Yet
  Scripture at the beginning of that narrative does not say, three men appeared to him, but, "The Lord appeared to him." And then, setting
  forth in due order after what manner the Lord appeared to him, it has
  added the account of the three men, whom Abraham invites to his
  hospitality in the plural number, and afterwards speaks to them in the
  singular number as one; and as one He promises him a son by Sara, viz.
  the one whom the Scripture calls Lord, as in the beginning of the same
  narrative, "The Lord," it says, "appeared to Abraham." He invites them
  then, and washes their feet, and leads them forth at their departure,
  as though they were men; but he speaks as with the Lord God, whether
  when a son is promised to him, or when the destruction is shown to him
  that was impending over Sodom.


Genesis 18 is taken seriously by many Trinitarians to prove that Trinity is mentioned in the Old Testament. The language used in Genesis 18 is vague and subjected to many interpretations. This has to be one of the most debated passages of all time. Non-Trinitarians and even some Trinitarians understand this as "God and two angels". However, as a Trinitarian, I prefer to believe that it was The Trinity who visited Abraham.

Explanation why the two men were not merely angels

Genesis 19 called the two men as angels. However, as the language here is vague, it seems the two angels were also the Lord(or Lords). For they said "we are going to destroy this place" and then later the Bible says "God destroyed the cities".


  (Genesis 19, NIV)
  
  (1) The two angels arrived at Sodom in the evening, and Lot was sitting in
  the gateway of the city. When he saw them, he got up to meet them and
  bowed down with his face to the ground. 
  
  (12) The two men said to Lot, “Do you have anyone else here—sons-in-law,
  sons or daughters, or anyone else in the city who belongs to you? Get
  them out of here, (13) because we are going to destroy this place.
  The outcry to the Lord against its people is so great that he has sent
  us to destroy it.”
  
  (24) Then the Lord rained down burning sulfur on Sodom and
  Gomorrah—from the Lord out of the heavens.
  
  (29) So when God destroyed the cities of the plain, he remembered
  Abraham, and he brought Lot out of the catastrophe that overthrew the
  cities where Lot had lived.