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The Antarctic ice sheet has been losing mass over past decades through the accelerated flow of its glaciers, conditioned by ocean temperature and bed topography. Glaciers retreating along retrograde slopes (that is, the bed elevation drops in the inland direction) are potentially unstable, while subglacial ridges slow down the glacial retreat. Despite major advances in the mapping of subglacial bed topography, significant sectors of Antarctica remain poorly resolved and critical spatial details are missing. Here we present a novel, high-resolution and physically based description of Antarctic bed topography using mass conservation. Our results reveal previously unknown basal features with major implications for glacier response to climate change. For example, glaciers flowing across the Transantarctic Mountains are protected by broad, stabilizing ridges. Conversely, in the marine basin of Wilkes Land, East Antarctica, we find retrograde slopes along Ninnis and Denman glaciers, with stabilizing slopes beneath Moscow University, Totten and Lambert glacier system, despite corrections in bed elevation of up to 1 km for the latter. This transformative description of bed topography redefines the high- and lower-risk sectors for rapid sea level rise from Antarctica; it will also significantly impact model projections of sea level rise from Antarctica in the coming centuries.