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Biofilms inhabit a range of environments, such as dental plaques or soil micropores, often characterized by intricate, non-even surfaces. However, the impact of surface irregularities on the population dynamics of biofilms remains elusive as most biofilm experiments are conducted on flat surfaces. Here, we show that the shape of the surface on which a biofilm grows influences genetic drift and selection within the biofilm. We culture E. coli biofilms in micro-wells with an undulating bottom surface and observe the emergence of clonal sectors whose size corresponds to that of the undulations, despite no physical barrier separating different areas of the biofilm. The sectors are remarkably stable over time and do not invade each other; we attribute this stability to the characteristics of the velocity field within the growing biofilm, which hinders mixing and clonal expansion. A microscopically-detailed computer model fully reproduces these findings and highlights the role of mechanical (physical) interactions such as adhesion and friction in microbial evolution. The model also predicts clonal expansion to be severely limited even for clones with a significant growth advantage - a finding which we subsequently confirm experimentally using a mixture of antibiotic-sensitive and antibiotic-resistant mutants in the presence of sub-lethal concentrations of the antibiotic rifampicin. The strong suppression of selection contrasts sharply with the behavior seen in bacterial colonies on agar commonly used to study range expansion and evolution in biofilms. Our results show that biofilm population dynamics can be controlled by patterning the surface, and demonstrate how a better understanding of the physics of bacterial growth can pave the way for new strategies in steering microbial evolution., Comment: 15 pages and 4 figures (main text), 13 pages and 6 figures (supplementary material)