K-flow barriers with boundary tangency comparison and uniform height bound

The reference collar in [F3] is a smooth compact q-family, and [F4] gives $K=0$ on S and ${\mathrm{\nabla }}_{{\bar{\nu }}_q}^{(q)}K=\kappa {\bar{n}}_q$. At $c=0$ the variable-time graph differential on S is the identity, because its time is zero there and K vanishes. It therefore has exactly the reference tangent plane and future normal. For a reference unit radial curve x(rho), differentiating ${\varphi }_{\mp M\rho }^{K}x(\rho )$ twice at zero changes its covariant acceleration by $\mp 2M{\mathrm{\nabla }}_{{\bar{\nu }}_q}K=\mp 2\kappa M{\bar{n}}_q$. Tangential directions in TS have unchanged second form because the hypersurfaces have the same boundary embedding. Since $H=-{\sum }_i{g}_q({\bar{n}}_q,{\mathrm{\nabla }}_{e_i}{e}_i)$ at the shared first jet, the radial change gives precisely the displayed $\mp 2\kappa M$ difference.

The family in the Statement meets all hypotheses of [F6], including actual physical exterior-section maximality, the explicit two-sided original-metric far-end access premise, the selected single-primitive family and uniform end-estimate requirements. The union $C$ of the reference outer boundaries ${\rho }_q={\rho }_0$, for $q\in [\delta ,1]$, is the image of $[\delta ,1]\times S$ under the smooth boundary-distance map at radius ${\rho }_0$ from [F3]. Since $S$ is compact, [F16]–[F18] make $C$ compact; the common collar bounds keep it away from $S$. For a fixed positive period $P$ of the axial flow, its saturation is the continuous image of $[0,P]\times C$, hence is compact by [F16]–[F18]. Its smooth stationary projection is compact in the original section by [F16], and its continuous stationary time coordinate is bounded by the extreme-value part of [F16]. For $F_q(t,x)=t-u_q(x)$, the identities in [F7] give $F_q({\varphi }_s^{K}x)=s+F_q({\varphi }_{{\mathrm{\Omega }}_{H}s}^{Z}x)$. At the unique intersection from [F5], this is zero. The second term is bounded uniformly by [F6], the compact projection and periodicity. Thus $\mathrm{\mid }{u}_q^K\mathrm{\mid }\le B_{\delta }$ on every outer reference boundary, for a common constant.

The boundary mean curvature is smooth on the compact parameter space $[0,1]\times S$ from [F3], so [F16]–[F18] give its finite uniform bound used to choose M. The chosen M gives a strict margin larger than one at S for all q. Compactness and the common smooth collar bounds permit one smaller rho0 on which both $c=0$ barriers remain embedded and spacelike and have the strict signs. Interior injectivity is also immediate from the reference K-flow product in [F3], and no finite orbit from an interior point can reach a zero of K. The boundary immersion has the reference tangent by step 1.1. Every translated barrier is the isometric image $B_c^{\pm }={\varphi }_c^K(B_0^{\pm })$ in the supplied complete collar saturation, fixing S. The connected isometric flow preserves future orientation, hence spacelikeness and the asserted signs for every real c.

Fix q. The closed reference collar and its outer boundary are nonempty compact images of $S\times [0,{\rho }_0]$ and $S$, respectively, under the smooth boundary-distance map from [F3], by [F16]–[F18]. Smooth extension in [F5] makes $f=u_q^K+M\rho$ continuous there, so [F16] gives a finite maximum c* and an attained outer-boundary maximum. If c* exceeded its outer-boundary maximum, a maximizing point would lie in the interior or on S. The plus barrier with height $c\ast -M\rho$ lies on the future side of the candidate, by [F13], and touches it. At an interior contact the height difference has zero differential, so the surfaces are tangent. At a boundary contact $u_q^K(p)=c\ast$, so [F12] supplies tangency and the common exterior side independently of height derivatives. [F14], using the positive-symbol and half-domain facts [F10]–[F11], would imply $H(B_{c\ast }^{+})\ge 0$, contrary to step 2.1. Therefore $u_q^K+M\rho \le {\max }_{\rho ={\rho }_0}{u}_q^K+M{\rho }_0$. All local comparison times are finite because c* and the individual smooth boundary heights are finite.

By [F16] on the same nonempty compact sets from step 3.1, the continuous function $u_q^K-M\rho$ attains a finite minimum $c_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and an outer-boundary minimum. Apply the reversed argument to this minimum. If it were less than its outer-boundary minimum, the minus barrier with height $c_{\mathrm{m}\mathrm{i}\mathrm{n}}+M\rho$ would touch the maximal candidate from the past at an interior or boundary point. The same tangency statements apply, and [F14] would require $H(B_{c_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{-})\le 0$, contrary to its strict positive sign. Hence $u_q^K-M\rho \ge {\min }_{\rho ={\rho }_0}{u}_q^K-M{\rho }_0$. Combining this with step 3.1 and the outer bound in step 1.2 gives $\mathrm{\mid }{u}_q^K\mathrm{\mid }\le B_{\delta }+M{\rho }_0$. This constant is independent of q on the fixed positive interval.