  According to conformal cyclic cosmology, a two-surface (Penrose 2-surface)
must exist between successive conformal aeons. To understand the quantum
gravitational effects of the Penrose 2-surface, a few higher dimensional
solutions are required. The higher dimensional metric for the Penrose 2-surface
is obtained in the current work. The evolution of the conformal factor is
discussed in this work.This article discusses the importance of higher
dimensional solutions in understanding the final stages of the universe,
particularly in the context of conformal cyclic cosmology. The article explores
the behaviour of de Sitter space via eternal inflation, with the aim of
providing a more complete understanding of the evolution of the universe and
the role that higher dimensions may play in its final stages. The article
concludes that without the availability of higher dimensions, the conformal
birth of a new aeon will not be possible, and that the use of only three
spatial dimensions in the final 2-surface of an aeon will fall short in
accounting for the influence of quantum gravity. The article also presents a
5-dimensional metric that describes the behaviour of 5-dimensional spacetime,
along with the corresponding 5-dimensional Friedmann-Robertson-Walker equations
of the conformal surface in the late-time evolution of the universe. The
results of this work have important implications for our understanding of the
nature of space and time and how they behave in extreme conditions, such as the
final stages of the universe.
