Why deep ocean didn't freeze during snowball Earth? I know that at least twice in its history Earth was totally frozen. I also know that the last time it happened there were already life on Earth, and it survived. In deep water, under the ice, which was transparent enough for light to penetrate it and support life. 

My question is, why didn't all the water freeze? AFAIK Earth was frozen for 10 million years, so I think it was enough time. Something must have kept deep water from freezing, what was it? Geothermal warmth? Underwater volcanoes? Both seem to be too weak sources of energy for keeping deep ocean liquid.

The same about Antarctica. I know there was a lake found deep under thick ice, why didn't it freeze?
At least four things combined to prevent solid freeze-up during Ice/Slush Ball Earth periods:   


It takes a lower temperature to freeze water under pressure. Deep sea pressures are enormous.   
Ice floats. If it sank and new ice kept forming at surface, the seas might fill with ice.   
The surface freezes first and acts as an approx zero-degree C barrier, slowing the progress of the ice penetration.   
Thermal heat from the ocean floor at hotspots, island arcs, subduction zones, and mid-oceanic rifts is substantial, especially locally.