Abstract
The kernel of the evolution equation is calculated in a (2+1)-dimensional Euclidean spacetime, which is a product of the compact two-dimensional manifold and the closed time coordinate. The exact solution for the trace of this evolution kernel axiomatically gives, via the proper time integral with a lower bound, a dimensionless functional of a dimensionless variable. The derivative of the obtained universal sum and its asymptotics are analyzed. These geometrical results serve as a mathematical foundation for the new thermal theory of two-dimensional systems and surfaces, with the input physical observables of velocity of sound, inter-atomic distance and thermodynamic temperature. The behaviour of the specific heat of two-dimensional materials, at the quasi-low temperature regime, is predicted to be the universal law of a cubic power of temperature.
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