We prove the existence and uniqueness of the Robin heat kernel on compact Riemannian manifolds with smooth boundary for Robin parameter α ∈ R R, expressed as a spectral expansion in terms of Robin eigenvalues and eigenfunctions. For the non-negative parameter regime (α ≥ 0 0), we present a direct proof based on trace Sobolev inequalities and eigenfunction estimates. The case of negative parameters (α > 0 >0) requires novel analytical techniques to handle L ∞ L^ estimates of Robin eigenfunctions, addressing challenges not present in the non-negative case. Our result extends the classical Dirichlet and Neumann cases to the less-studied negative parameter regime.
Meng et al. (Thu,) studied this question.