Higher regularity of the free boundary in the parabolic Signorini problem

We show that the quotient of two caloric functions which vanish on a portion of an $H^{k+\alpha }$ regular slit is $H^{k+\alpha }$ at the slit, for $k \geq 2$\mathrm{.}Inthecase$k=1$,weshowthatthequotientisin$H^{1+\alpha}$ if the slit is assumed to be space-time $C^{1,\alpha }$ regular. This can be thought of as a parabolic analogue of a recent important result in [DSS14a], whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in [DGPT] with zero obstacle is $C^{\mathrm{\infty }}$ regular in space and time.