# Heat Equation on Manifold

Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold.

So is there any analogy to heat equation or wave equation on manifold? And what book is recommended for beginner to read about this field.

Thank you.

• Heat type equation on manifold is a huge success. As a starter you can the book "Laplacian on a Riemannian manifold" by S. Rosenberg.
– user99914
Dec 1 '14 at 11:30

The heat and wave equations have very nice analogous equations on Riemannian manifolds $(M,g)$. If the Laplace-Beltrami operator is given by: $$\Delta_g = \text{div}_g \,\nabla_g$$ Then the heat and wave equations, respectively, are: $$\Delta_g u =\gamma_h\, \partial_t u$$ $$\Delta_g u = \gamma_w\, \partial_{tt} u$$ for constants $\gamma_h,\gamma_w$. Pleasantly, their solutions can be written in terms of the spectrum of $\Delta_g$: $$u(x,t) = \sum_i \alpha_i \exp(-\lambda_i t)\phi_i(x)$$ $$u(x,t) = \sum_j \alpha_j \exp\left(it\sqrt{\lambda_j}\,\right)\phi_j(x)$$ where $\Delta_g \phi_\ell = -\lambda_\ell\phi_\ell$ (the Helmholtz eigenvalue equation of $M$) and $\alpha_i$ depend on the initial conditions, for the heat and wave equations respectively.
You can also compute the kernels of the PDEs, i.e. $K_t(x,y,t)$ s.t. $$f(x,t) = \int_M K(x,y,t)f(y,0)dy$$ is a solution to the heat/wave equation, where the kernel is given by $$K(x,y,t) = \sum_j \phi_j(x)\phi_j(y) \exp( \xi_j t)$$ with $\xi_j = -\lambda_j$ for the heat equation and $\xi_j=i\sqrt{\lambda_j}$ for the wave equation.