Exponential energy decay on totally real boundary necks

A fixed finite-atlas embedding is obtained as follows. Choose a finite smooth partition ${\varphi }_i$ with compact supports in coordinate charts $x_i$, using finite Euclidean bumps and normalization, and set $F(x)=({\varphi }_i(x),{\varphi }_i(x)x_i(x){)}_i$. Equality of these coordinates recovers $x$ in a chart where ${\varphi }_i>0$, and vanishing of their derivatives recovers a zero tangent vector in that chart. Thus $F$ is injective and immersive, and compactness makes it an embedding. Derivatives of $F$ are bounded and its differential uniformly compares $g$ with the ambient Euclidean metric. Write $Y=F(X)$ and $M=F(L)$.

We construct a smooth involution $\rho$ on a neighborhood of $L$ in $X$, fixing $L$, whose differential is $+I$ on $TL$ and $-I$ on $J_{b}TL$. First the ambient normal map $(y,n)\mapsto y+n$ from the Euclidean orthogonal normal bundle of $Y$ has invertible differential along its zero section. By [F3] it has local smooth inverses. It is injective for a uniform small normal radius: otherwise collisions with normal vectors tending to zero have base points converging after a subsequence by compactness; equality of images forces the same base limit, contradicting the local inverse there. The same compactness argument preserves invertible differential on a uniform radius. Projection of these glued inverses to $Y$ gives a smooth ambient retraction $P$ near $Y$ with $dP$ the tangent projection along $Y$. Now on the vector bundle $N_y=dF(J_b{T}_{x}L)$ over $y=F(x)\in M$, consider $\mathrm{\Phi }(y,v)=P(y+v)\in Y$. At $v=0$ its derivative identifies $T_{y}M\oplus N_y$ with $T_{y}Y$, since $L$ is totally real. The identical local-inverse and compactness argument gives a diffeomorphism from a uniform small symmetric bundle neighborhood onto a neighborhood of $M$ in $Y$. Conjugating $(y,v)\mapsto (y,-v)$ by this diffeomorphism and then by $F$ defines $\rho$. It is an involution, fixes $L$, and has the claimed differential. On a smaller fixed neighborhood all its first and second derivatives, and its inverse differential, are bounded.

For small enough $\epsilon$, the entire map lies in that fixed neighborhood: each path $\tau \mapsto u(\sigma ,\tau )$ begins on $L$ and has length at most $T\epsilon$. Define on the doubled angular circle of period $2T$ $\tilde{u}(\sigma ,\tau )=\{\begin{array}{ll}u(\sigma ,\tau ),& 0\le \tau \le T,\\ \rho (u(\sigma ,-\tau )),& -T\le \tau \le 0,\end{array}$ identifying $-T$ and $T$. This is $C^1$ across both seams. Values agree because $\rho$ fixes $L$. Tangential derivatives agree because $u_{\sigma }\in TL$ there, so $d\rho u_{\sigma }=u_{\sigma }$. The equation gives $u_{\tau }=J_b{u}_{\sigma }\in J_{b}TL$, so the normal reflected derivative $-d\rho u_{\tau }$ equals $u_{\tau }$. These identities hold along the whole seam and may also be differentiated in $\sigma$. Thus all tangential derivatives of the values and normal first derivatives match. The doubled map is smooth on each closed side.

Put $V=F\circ \tilde{u}$ and let $H=\mathrm{\Delta }V$ weakly on the doubled cylinder. There is no line measure in this Laplacian because $V$ and its first derivatives match across the seams; integrating by parts on both sides verifies cancellation. On the original half, the composition formula and [F1] bound $\mathrm{\mid }H\mathrm{\mid }\le C(\mathrm{\mid }du{\mathrm{\mid }}_g^2+\kappa \mathrm{\mid }du{\mathrm{\mid }}_g)$. On the reflected half the composition rule for $F\circ \rho$ gives the same bound, with a larger fixed constant from its second derivatives. The equation gives $\mathrm{\mid }du{\mathrm{\mid }}_g\le C\mathrm{\mid }{u}_{\tau }{\mathrm{\mid }}_g$, and the embedding and reflected inverse differential compare this with $\mathrm{\mid }{V}_{\tau }\mathrm{\mid }$ on either side. Hence $\mathrm{\mid }H\mathrm{\mid }\le C_0\epsilon \mathrm{\mid }{V}_{\tau }\mathrm{\mid }$ almost everywhere on the entire doubled cylinder. This argument does not assume a globally smooth reflected almost complex coefficient across a seam.

Set $w=V_{\tau }$ and $e(\sigma )=\frac{1}{2}{\int }_{-T}^T\mathrm{\mid }w{\mathrm{\mid }}^{2}d\tau$. The function $w$ is continuous and piecewise smooth, its angular derivative is an ordinary bounded piecewise function locally, and it has zero angular mean. Moreover $\mathrm{\Delta }w={\mathrm{\partial }}_{\tau }H$ as distributions. Tangential differentiation of the piecewise formulas gives $e^{\mathrm{\prime }\mathrm{\prime }}={\int }_{-T}^T(\mathrm{\mid }{w}_{\sigma }{\mathrm{\mid }}^2+\mathrm{\mid }{w}_{\tau }{\mathrm{\mid }}^2-⟨w_{\tau },H⟩)d\tau \mathrm{.}$ For completeness the last integration by parts is distributional: $w$ is periodic $H^1$ in its angular variable and $H\in L^2$, so convolution on that circle proves $⟨{\mathrm{\partial }}_{\tau }H,w⟩=-\int ⟨H,w_{\tau }⟩$ by passage to the $L^2$ limit. Thus any jumps of $H$ are included, and no angular endpoint term was omitted. The matching tangential derivatives from step 3.1 justify differentiating the integral twice in $\sigma$.

Young's inequality and [F2] give, for $\lambda =(\pi \mathrm{/}T{)}^2$, $e^{\mathrm{\prime }\mathrm{\prime }}\ge \frac{1}{2}\int \mathrm{\mid }{w}_{\tau }{\mathrm{\mid }}^2-\frac{1}{2}{C}_0^2{\epsilon }^2\int \mathrm{\mid }w{\mathrm{\mid }}^2\ge \frac{1}{2}(\lambda -C_0^2{\epsilon }^2)\int \mathrm{\mid }w{\mathrm{\mid }}^2\mathrm{.}$ Decrease $\epsilon$ so that $C_0^2{\epsilon }^2\le \lambda \mathrm{/}2$ and set $c=\sqrt{\lambda \mathrm{/}2}$. Then $e^{\mathrm{\prime }\mathrm{\prime }}\ge c^{2}e$. The barrier $h=e(a)e^{-c(\sigma -a)}+e(b)e^{-c(b-\sigma )}$ has $h^{\mathrm{\prime }\mathrm{\prime }}=c^{2}h$ and dominates the endpoint values of $e$. A positive interior maximum of $e-h$ would contradict $(e-h{)}^{\mathrm{\prime }\mathrm{\prime }}\ge c^2(e-h)$, so $e\le h$. Its integral on $[a+R,b-R]$ is at most $c^{-1}(e(a)+e(b))e^{-cR}$.

The original energy is bounded by $C\int e(\sigma )d\sigma$, by the original equation and embedding norm equivalence. The small derivative hypothesis and the fixed reflected derivative bound give $e(a)+e(b)\le C{\epsilon }^2$. Substitution in step 6.1 proves the asserted central energy estimate with a uniform constant. This proves boundary neck decay using the actual totally real boundary geometry. It does not address two different submanifolds or construct a bubble-tree decomposition.