Sharpness-aware minimization (SAM) reports improving domain generalization byreducing the loss surface curvature in the parameter space. However,generalization during fine-tuning is often more dependent on thetransferability of representations in the function space. Trust-regionmethods (TR) target this goal by regularizing representation curvature to reducecatastrophic forgetting of pre-trained task-agnostic information while adoptingtask-specific skills. We consider unifying these strategies for low curvature inboth parameter space and function space to improve out-of-domain (OOD)generalization. We propose Trust Region Aware Minimization (TRAM), aSAM algorithm fine-tuning for low parameter sharpness and smooth, informativerepresentations preserving pre-trained structure. TRAM uses a trust region boundto inform the SAM adversarial neighborhood, introducing an awareness of functioncurvature within optimization for flatter minima. We empirically validate TRAMin vision (cross-dataset adaptation) and text (OOD language modeling, zero-shotcross-lingual transfer) tasks where robust domain transfer and representationgenerality are critical. TRAM outperforms SAM- and TR-based optimization acrossall tasks, notably surpassing competing methods for hard transfer betweenanticorrelated domains. TRAM establishes a novel standard infine-tuning for domain-generalizable models with minimal additional computationover previous sharpness-aware methods.