Exponential energy decay on small-gradient almost complex cylinder necks

Choose a finite chart cover and smooth subordinate functions ${\varphi }_i$ with compact supports in those charts and sum one, obtained by Euclidean bumps and normalization. The map $F(x)=({\varphi }_i(x),{\varphi }_i(x)x_i(x){)}_i$ is a smooth embedding: at an index where ${\varphi }_i>0$, equality of the displayed coordinates recovers the original chart point, and vanishing of all their derivatives recovers a zero tangent vector. Compactness makes it a homeomorphism onto its image and makes its differential uniformly comparable with the metric $g$. Its first and second covariant derivatives have uniform bounds. This fixes $F$ independently of the cylinder and its map.

Let $H=\mathrm{\Delta }V$, taking the ordinary flat source Laplacian componentwise. The composition rule and [F1] give $H=dF({\mathrm{\nabla }}_{\sigma }{u}_{\sigma }+{\mathrm{\nabla }}_{\tau }{u}_{\tau })+(\mathrm{\nabla }dF)(u_{\sigma },u_{\sigma })+(\mathrm{\nabla }dF)(u_{\tau },u_{\tau })\mathrm{.}$ Uniform coefficient and embedding bounds therefore give $\mathrm{\mid }H\mathrm{\mid }\le C_0(\mathrm{\mid }du{\mathrm{\mid }}_g+\kappa )\mathrm{\mid }du{\mathrm{\mid }}_g$. The original equation and the bound for $J$ give $\mathrm{\mid }du{\mathrm{\mid }}_g\le C_1\mathrm{\mid }{u}_{\tau }{\mathrm{\mid }}_g\le C_2\mathrm{\mid }{V}_{\tau }\mathrm{\mid }$. After imposing the stated smallness, $\mathrm{\mid }H\mathrm{\mid }\le C_3\epsilon \mathrm{\mid }{V}_{\tau }\mathrm{\mid }\mathrm{.}$ This is a pointwise estimate on actual maps into $X$; no ambient extension of $J$ or model moduli equation is assumed.

Put $w=V_{\tau }$. It has zero angular mean because $V$ is periodic, and $\mathrm{\Delta }w={\mathrm{\partial }}_{\tau }H$ because derivatives of a smooth Euclidean map commute. Twice differentiating $e=\frac{1}{2}\int \mathrm{\mid }w{\mathrm{\mid }}^2$ and integrating by parts only in the periodic angular direction gives $e^{\mathrm{\prime }\mathrm{\prime }}={\int }_0^T(\mathrm{\mid }{w}_{\sigma }{\mathrm{\mid }}^2+\mathrm{\mid }{w}_{\tau }{\mathrm{\mid }}^2-⟨w_{\tau },H⟩)d\tau \mathrm{.}$ There are no endpoint terms in this integration. By step 2.1 and Young's inequality this is at least ${\int }_0^T\mathrm{\mid }{w}_{\sigma }{\mathrm{\mid }}^2+\frac{1}{2}\mathrm{\mid }{w}_{\tau }{\mathrm{\mid }}^2-\frac{1}{2}{C}_3^2{\epsilon }^2\mathrm{\mid }w{\mathrm{\mid }}^{2}d\tau \mathrm{.}$ Let $\lambda =(2\pi \mathrm{/}T{)}^2$ and choose $\epsilon >0$ so that $C_3^2{\epsilon }^2\le \lambda \mathrm{/}2$. Then [F2] gives $e^{\mathrm{\prime }\mathrm{\prime }}\ge (\lambda \mathrm{/}4)\int \mathrm{\mid }w{\mathrm{\mid }}^2=(\lambda \mathrm{/}2)e$. Set $c=\sqrt{\lambda \mathrm{/}2}$.

Let $h(\sigma )=e(a)e^{-c(\sigma -a)}+e(b)e^{-c(b-\sigma )}$. It satisfies $h^{\mathrm{\prime }\mathrm{\prime }}-c^{2}h=0$ and dominates $e$ at both endpoints. If $e-h$ had a positive value, it would have a positive interior maximum; there its second derivative is nonpositive, contrary to $(e-h{)}^{\mathrm{\prime }\mathrm{\prime }}\ge c^2(e-h)>0$. Thus $e\le h$ throughout the cylinder, giving the claimed pointwise exponential bound. The argument also covers vanishing endpoint values, since $e$ is nonnegative and smooth.

The equation and uniform metric comparability give $E_g(u;[a+R,b-R]\times S_T^1)\le C_4{\int }_{a+R}^{b-R}e(\sigma )d\sigma \mathrm{.}$ Integrating the two exponentials from step 4.1 bounds the last integral by $c^{-1}(e(a)+e(b))e^{-cR}$. Finally $\mathrm{\mid }{V}_{\tau }\mathrm{\mid }\le C\mathrm{\mid }{u}_{\tau }{\mathrm{\mid }}_g\le C\epsilon$, so $e(a)+e(b)\le CT{\epsilon }^2$. This proves all assertions, including uniform decay of central neck energy. A boundary neck has angular endpoint terms and is not covered by the periodic argument.