This work describes a geometrical approach to critical phenomena in bounded domains. A metric is introduced in a bounded system, forcing the curvature of the system to be uniform. This leads to the Yamabe equation, which is modified into the fractional Yamabe equations for models with nonvanishing anomalous dimension. A way to define the fractional Laplacian in bounded domains and solve the corresponding equation is presented.
From its solution, correlation functions for the fields can be obtained. These predictions are then tested against numerical simulations for various models, both at the upper critical dimension (four-dimensional Ising model) and below (three-dimensional XY model and percolation), finding accurate agreement and extracting values for the respective critical exponents η. In two dimensions, it is shown how this theory recovers results from boundary conformal field theory.
