Totten Glacier, the primary outlet of the Aurora Subglacial Basin, has the largest thinning rate in East Antarctica. Thinning may be driven by enhanced basal melting due to ocean processes, modulated by polynya activity. Warm modified Circumpolar Deep Water, which has been linked to glacier retreat in West Antarctica, has been observed in summer and winter on the nearby continental shelf beneath 400 to 500 m of cool Antarctic Surface Water. Here we derive the bathymetry of the sea floor in the region from gravity and magnetics data as well as ice-thickness measurements. We identify entrances to the ice-shelf cavity below depths of 400 to 500 m that could allow intrusions of warm water if the vertical structure of inflow is similar to nearby observations. Radar sounding reveals a previously unknown inland trough that connects the main ice-shelf cavity to the ocean. If thinning trends continue, a larger water body over the trough could potentially allow more warm water into the cavity, which may, eventually, lead to destabilization of the low-lying region between Totten Glacier and the similarly deep glacier flowing into the Reynolds Trough. We estimate that at least 3.5 m of eustatic sea level potential drains through Totten Glacier, so coastal processes in this area could have global consequences.