The Robin heat kernel and its expansion via Robin eigenfunctions

Abstract: We prove the existence and uniqueness of the Robin heat kernel on compact Riemannian manifolds with smooth boundary for Robin parameter $\alpha\in\mathbb{R}$, expressed as a spectral expansion in terms of Robin eigenvalues and eigenfunctions. For the non-negative parameter regime ($\alpha\ge 0$), we present a direct proof based on trace Sobolev inequalities and eigenfunction estimates. The case of negative parameters ($\alpha<0$) requires novel analytical techniques to handle $L^\infty$ estimates of Robin eigenfunctions, addressing challenges not present in the non-negative case. Our result extends the the classical Dirichlet and Neumann cases to the less-studied negative parameter regime.