We present systematic experimental studies of the temperature dependence of continuous wave and time-resolved photoluminescence spectroscopy in self-assembled InAs/GaAs nanostructures as the shape of quantum structures evolved from dot to ring. The carrier dynamics show strong dependence on the geometrical shape of the nanostructures under investigation. An increase in photoluminescence decay time of the excited and ground states is observed as the shape of nanostructures changed from dot, volcano, to ring. It is attributed to the carrier thermalization between the dark and ground states. The photoluminescence excitation spectra of the quantum rings reveal resonances related to the dark states. A rate equation model is proposed to interpret the observed carrier dynamics.