As heat kernel analysis quantifies the space-time evolution of temperature in a given object, it is a powerful tool in studying geometric properties of domains on manifolds. For instance, De Giorgi considered the heat semigroup convolved with the Gauss-Weierstrass kernel

\[

P_t(\chi_D)(x)=\frac{1}{(2\pi t)^{n/2}}\int_De^{-\frac{|x-y|^2}{2t}}dy,

\]

and realized the existence of the limit, which defines the perimeter of domain D.

\[

\lim_{t\to0}\sqrt{\frac{\pi}{t}}\int_{D^c}P_t(\chi_{D})(x)dx=P(D).

\]

The intuition behind this is simple. In short time, the heat that passes through domain D is proportional to its perimeter. Van de Berg-Le Gall obtained a more refined result by deriving the small time expansion of the heat content

\[

Q_D(t):=\int_{D^c}P_t(\chi_{D})(x)dx

\]

which captures the information of volume, perimeter, and average mean curvature of the boundary of the domain.

\[

Q_D(t)=|D|-\sqrt{\frac{2t}{\pi}}|\partial D|+\frac{t}{4}\int_{\partial D}H_D(x)ds+o(t).

\]

In a current project with J. Tyson, we obtain small time expansion of heat content of a domain D with smooth boundary in the Heisenberg group.

\[

Q_D(t)=\mathrm{Vol}(D)-\sqrt{\frac{2t}{\pi}}\sigma_{\mathcal{H}}(\partial D)+\frac{t}{4}\int_{\partial D}H_{\mathcal{H}}(x)\,d\sigma_{\mathcal{H}}(x)+o(t).

\]

\[

P_t(\chi_D)(x)=\frac{1}{(2\pi t)^{n/2}}\int_De^{-\frac{|x-y|^2}{2t}}dy,

\]

and realized the existence of the limit, which defines the perimeter of domain D.

\[

\lim_{t\to0}\sqrt{\frac{\pi}{t}}\int_{D^c}P_t(\chi_{D})(x)dx=P(D).

\]

The intuition behind this is simple. In short time, the heat that passes through domain D is proportional to its perimeter. Van de Berg-Le Gall obtained a more refined result by deriving the small time expansion of the heat content

\[

Q_D(t):=\int_{D^c}P_t(\chi_{D})(x)dx

\]

which captures the information of volume, perimeter, and average mean curvature of the boundary of the domain.

\[

Q_D(t)=|D|-\sqrt{\frac{2t}{\pi}}|\partial D|+\frac{t}{4}\int_{\partial D}H_D(x)ds+o(t).

\]

In a current project with J. Tyson, we obtain small time expansion of heat content of a domain D with smooth boundary in the Heisenberg group.

\[

Q_D(t)=\mathrm{Vol}(D)-\sqrt{\frac{2t}{\pi}}\sigma_{\mathcal{H}}(\partial D)+\frac{t}{4}\int_{\partial D}H_{\mathcal{H}}(x)\,d\sigma_{\mathcal{H}}(x)+o(t).

\]