# Heat kernel on hyperbolic space of variable curvature

I am working with the heat kernel on the hyperbolic space explicitly (as you may guess by my previous questions) and I got the desired results when the curvature is $$-\kappa=-1$$. Now I am trying to do the same for a fixed but arbitrary curvature $$-\kappa<0$$, so I need to generalize the explicit formulas for the heat kernel that I got in $$\mathbb{H}^n(1)$$ (see ''The Heat Kernel on Hyperbolic Space'' or ''Heat kernel bounds on hyperbolic space and Kleinian space'') to a more general $$\mathbb{H}^n(\kappa)$$.

I achieve to get a general formula when $$n$$ is odd, that is $$p_{n}(\rho, t)=\frac{(-1)^{m}}{2^{m} \pi^{m}} \frac{1}{(4 \pi t)^{\frac{1}{2}}}\left(\frac{\kappa}{\sinh (\kappa\rho)} \frac{\partial}{\partial \rho}\right)^{m} e^{-\kappa^2m^{2} t-\frac{\rho^{2}}{4 t}}$$ with $$n=2m +1$$, but I did it by just adding $$\kappa$$ in the formula and then cheking that it satisfies the equation, that is now $$\frac{\partial^2}{\partial \rho^2}p_{n}(\rho, t)+(n-1)\kappa \coth(\kappa \rho)\frac{\partial}{\partial \rho}p_{n}(\rho, t)-\frac{\partial}{\partial t}p_{n}(\rho, t).$$ When I try to do the same to even $$n$$, it fails because I can not compute the derivatives so easily.

Could someone help me find a general formula when $$n=2m+2$$? I have some intuition but I can not check if they are, in fact, the fundamental solution that I want.

The sectional curvature scales like the inverse of the metric. So fixing a coordinate system on $$\mathbb{H}^n(1)$$, with metric $$g$$, the scaled metric $$\kappa^{-1} g$$ has sectional curvature $$-\kappa$$.

If $$u(t,x)$$ solves the heat equation you have $$u_t = \Delta_g u \iff \kappa u_t = \kappa \Delta_g u \iff \kappa u_t = \Delta_{\kappa^{-1} g} u$$ and so $$u(\kappa t,x)$$ solves the heat equation for the $$-\kappa$$ curvature.

The function $$\rho$$ being the geodesic distance scales like $$\kappa^{1/2}$$: that is $$\rho_g = \kappa^{1/2} \rho_{\kappa^{-1} g}$$.

And the volume form scales like $$\mathrm{dvol}_g = \kappa^{n/2} \mathrm{dvol}_{\kappa^{-1} g}$$.

This tells you that if you write $$\tilde{p}$$ for the heat kernel when the section curvature equals $$-\kappa$$, and $$p$$ for the heat kernel when the sectional curvature equals $$-1$$, you should have

$$\tilde{p}_n(\rho,t) = \kappa^{n/2} p_n(\kappa^{1/2}\rho, \kappa t)$$

So first: your formula for the odd dimensional case is wrong. Where you have $$\kappa$$ you should have $$\kappa^{1/2}$$ instead (unless you are actually looking at the case where sectional curvature equals $$-\kappa^2$$). The correct formula should be

$$\tilde{p}_{n}(\rho, t)=\frac{(-1)^{m}}{2^{m} \pi^{m}} \frac{1}{(4 \pi t)^{\frac{1}{2}}}\left(\frac{\kappa^{1/2}}{\sinh (\kappa^{1/2}\rho)} \frac{\partial}{\partial \rho}\right)^{m} e^{-\kappa m^{2} t-\frac{\rho^{2}}{4 t}}$$

In the even case (n = 2m+2) you should have

$$\tilde{p}_n(\rho, t) = \frac{(-1)^m}{2^{m+5/2} \pi^{m+3/2}} \kappa^{-1/2} t^{-3/2} e^{-\frac{(2m+1)^2}{4} \kappa t} \left( \frac{\kappa^{1/2}}{\sinh \kappa^{1/2}\rho} \partial_\rho \right)^m \int_{\kappa^{1/2}\rho}^\infty \frac{s e^{- s^2/(4\kappa t)}}{(\cosh s - \cosh \kappa^{1/2}\rho)^{1/2}} ds$$

• You're right. I was confused between $\kappa$ and $\kappa^2$. And the answer is very clear and helpful. Thank you. Nov 23, 2021 at 18:00