Heat-kernel asymptotics and geometric coefficients

The heat kernel propagates an initial temperature by integration against volume. Its spectral expansion weights each eigenfunction pair by the exponential of minus time times its eigenvalue. Integrating the diagonal uses orthonormality to produce the sum of those exponentials, counting multiplicity. The sum converges for every positive time on a compact smooth manifold. A constant zero mode contributes one at every time, even though high-frequency behavior dominates the short-time limit.

The full trace approaches the number of connected components as time tends to infinity, because all positive modes decay. At short time, by contrast, many high modes contribute, and a finite spectral truncation cannot reproduce the complete asymptotic uniformly down to zero time. One must distinguish a mode experiment at a fixed positive time from an infinite-spectrum short-time expansion. Their limiting operations occur in a particular order and cannot be exchanged without estimates.

For a closed d-dimensional manifold, the leading bracket coefficient is volume and the coefficient multiplying time is one sixth of the integrated scalar curvature. Higher terms involve curvature contractions and derivatives. These are scalar-function coefficients: changing to a form Laplacian, adding a potential, or changing an operator connection changes the formulas. Integrated coefficients are spectral invariants, but a single integrated curvature number does not determine the pointwise curvature function.

An asymptotic expansion means that subtracting finitely many terms leaves an error of the stated smaller order as time approaches zero. It need not be a convergent power series for a fixed positive time. On a closed surface the integrated scalar curvature equals four pi times the Euler characteristic, so the constant heat coefficient is the Euler characteristic divided by six. The ever-present zero mode is already included in the full spectral sum; it should not be added again to that geometric constant.

For a smooth compact manifold with smooth boundary, Dirichlet or Neumann conditions add a term inside the bracket equal to minus or plus one half of square root pi times boundary area times square root time. Subsequent coefficients include boundary curvature contributions as well as interior curvature. In dimension one, an interval has two boundary points, so this correction gives a constant of negative one half for Dirichlet and positive one half for Neumann. The different constants are not evidence for different interior scalar curvature.

For polygonal domains, corner angles contribute to later terms and a smooth-boundary formula cannot simply be reused. Likewise, removing the zero eigenspace changes the trace by its finite dimension and therefore changes its constant term, although the leading volume singularity is unchanged. When comparing heat data, identify the boundary realization and whether the trace is full or reduced. Those choices determine which coefficient can legitimately be interpreted as an audible geometric quantity.