Can a variac be used to maintain constant voltage?

That would be good application. They also bring supply voltage to regulation range of electronics in case of wide variations (constantly too low or too high).

Variacs are not specifically designed for audio applications, but rather for industrial, and general electronic testing. They operate very much like a large potentiometer, only- using inductance, rather than resistance, to divide voltage. Further- like a pot; they can only reduce voltages and cannot increase what is presented at their input, or maintain a set voltage, if a drop occurs at the line.

Actually, a Variac is a variable Autotransformer, which can be, and often is, designed to step voltage up as well as down. That would be done as shown in the first figure of the linked writeup, where the input voltage is placed across a segment of the winding that encompasses fewer than the full number of turns. Variacs that are designed for 120 volt single-phase inputs are commonly designed to output up to 140 volts, and sometimes more.

One other benefit they might provide, in addition to those Kijanki mentioned, is some reduction of high frequency noise that may be present on the incoming AC. However, as Rodman indicated Variacs are not specifically designed for audio applications, and an audio-oriented power conditioner (which in most cases won't regulate voltage) or a power regenerator (which will regulate voltage) can be expected to be a better alternative, albeit at a considerably higher price.

Regards,
-- Al