Materials based on minimal surface geometries have shown superior strength and stiffness at low densities, which makes them promising continuous-based material platforms for a variety of engineering applications. In this work, it is demonstrated how these mechanical properties can be complemented by dynamic functionalities resulting from robust topological guiding of elastic waves at interfaces that are incorporated into the consid- ered material platforms. Starting from the definition of Schwarz P minimal surface, geometric parametrizations are introduced that break spatial sym- metry by forming 1D dimerized and 2D hexagonal minimal surface-based materials. Breaking of spatial symmetries produces topologically non-trivial interfaces that support the localization of vibrational modes and the robust propagation of elastic waves along pre-defined paths. These dynamic prop- erties are predicted through numerical simulations and are illustrated by performing vibration and wave propagation experiments on additively manu- factured samples. The introduction of symmetry-breaking topological inter- faces through parametrizations that modify the geometry of periodic minimal surfaces suggests a new strategy to supplement the load-bearing properties of this class of materials with novel dynamic functionalities.