The Ehrhard-Borell inequality is a far-reaching refinement of the classical Brunn-Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Unlike in the classical Brunn-Minkowski theory, the equality cases in this inequality are far from evident from the known proofs. The equality cases are settled systematically in this paper. An essential ingredient of the proofs are the geometric and probabilistic properties of certain degenerate parabolic equations. The method developed here serves as a model for the investigation of equality cases in a broader class of geometric inequalities that are obtained by means of a maximum principle.