We investigate the rigidity and stability of eigenvalues of the heat kernel Laplacian on manifolds with nonnegative Ricci curvature. We show that the heat kernel Laplacian has discrete spectrum and its first eigenvalue is bounded below by the corresponding first eigenvalue of the Euclidean space, with equality holds if and only if the manifold splits off \(\mathbb {R}\) factors. Quantitatively, we show that if the \(k^{\text {th}}\) eigenvalue is close to that of the Euclidean space, then the manifold is close to a product of \(\mathbb {R}^k\) with some length space in the Gromov-Hausdorff distance. Building upon the continuity of eigenvalues, we show the converse is also true: if the manifold is close to a product of \(\mathbb {R}^k\) with some length space, then the \(k^{\text {th}}\) eigenvalue is close to that of the Euclidean space. Moreover, we characterize the necessary and sufficient conditions for asymptotic cones of a manifold to split off \(\mathbb {R}\) factors. As an application, if the \(n^{\text {th}}\) eigenvalue is close to that of the Euclidean space, we show that the manifold is diffeomorphic to \(\mathbb {R}^n\) and is isometric to \(\mathbb {R}^n\) when it is Ricci flat. We also prove a gap estimate for the \((n+1)^\text {th}\) weighted eigenvalue. To prove these results, we shall establish sharp Neumann-Poincaré and Log-Poincaré inequalities that are very useful in this setting.
