  By combined top- and backgating, we explore the correlation of
superconductivity with band filling and electron confinement at the
LaAlO$_3$-SrTiO$_3$ interface. We find that the top- and backgate voltages have
distinctly different effects on the superconducting critical temperature,
implying that the confining potential well has a profound effect on
superconductivity. We investigate the origin of this behavior by comparing the
gate-dependence of $T_c$ to the corresponding evolution of the band filling
with gate voltage. For several backgate voltages, we observe maximum $T_c$ to
consistently coincide with a kink in tuning the band filling for high topgate
voltage. Self-consistent Schr\"odinger-Poisson calculations relate this kink to
a Lifshitz transition of the second $d_{xy}$ subband. These results establish a
major role for confinement-induced subbands in the phase diagram of SrTiO$_3$
surface states, and establish gating as a means to control the relative energy
of these states.
