Totten Glacier, the primary outlet of the Aurora Subglacial Basin, has the largest thinning rate in East Antarctica^{1, 2}. Thinning may be driven by enhanced basal melting due to ocean processes³, modulated by polynya activity^{4, 5}. Warm modified Circumpolar Deep Water, which has been linked to glacier retreat in West Antarctica⁶, has been observed in summer and winter on the nearby continental shelf beneath 400 to 500 m of cool Antarctic Surface Water^{7, 8}. Here we derive the bathymetry of the sea floor in the region from gravity⁹ and magnetics¹⁰ data as well as ice-thickness measurements¹¹. 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.