Derivative concentration produces a finite-energy plane or half-plane bubble

Choose a flat source coordinate about $p$, centered there, with domain a disk or an upper half-disk. At the lower strip edge use the usual coordinate; at the upper edge use $z\mapsto i-z$, so both coordinates are orientation preserving and preserve the form of the equation. Write $f_n=\mathrm{\mid }d{u}_n{\mathrm{\mid }}_g$ in this coordinate and $M_n=f_n(p_n)\to \mathrm{\infty }$. After a tail, choose radii $r_n=\sqrt{\mathrm{\mid }{p}_n-p\mathrm{\mid }+M_n^{-1}}$ smaller than a fixed coordinate radius. Then $r_n\to 0$, $\mathrm{\mid }{p}_n-p\mathrm{\mid }\mathrm{/}{r}_n\to 0$, and $r_n{M}_n\to \mathrm{\infty }$. On the compact disk or half-disk of radius $r_n$ maximize $(r_n-\mathrm{\mid }z\mathrm{\mid })f_n(z)$, and let $q_n$ be a maximum point. Put ${\lambda }_n=f_n(q_n)$ and ${\delta }_n=(r_n-\mathrm{\mid }{q}_n\mathrm{\mid })\mathrm{/}2$. The maximum is at least $(r_n-\mathrm{\mid }{p}_n\mathrm{\mid })M_n$, so ${\lambda }_n{\delta }_n\to \mathrm{\infty }$, and in particular ${\lambda }_n\to \mathrm{\infty }$. On $B_{{\delta }_n}(q_n)$ intersected with the physical source half-plane, maximality gives $f_n\le 2{\lambda }_n$. All selected points tend to $p$.

If $p$ is interior, rescale by $z\mapsto q_n+z\mathrm{/}{\lambda }_n$. The domains contain disks about zero of radii ${\lambda }_n{\delta }_n\to \mathrm{\infty }$; the rescaled derivatives are at most two there and have norm one at zero. If $p$ is a boundary point, write $h_n={\lambda }_n\mathrm{Im}{q}_n\ge 0$. Pass either to $h_n\to \mathrm{\infty }$ or to a bounded sequence converging to $h\ge 0$. In the first case use the same center rescaling; its boundary recedes to height $-h_n$, so the limit domain is the plane. In the second case rescale about the boundary foot $\mathrm{Re}{q}_n$, obtaining upper half-plane domains on which the derivative bound holds on expanding disks centered at $i{h}_n$. Every fixed compact upper-half-plane set eventually lies in these disks, and the derivative norm is one at $i{h}_n$. The boundary values remain on the original fixed submanifold. In all cases the rescaled almost complex structures converge locally smoothly to $J_{\ast }(x)=J_{\mathrm{\infty }}(p,x)$: each positive source derivative gains a factor ${\lambda }_n^{-1}$ and the unscaled source points tend to $p$.

These rescaled maps have a subsequence converging smoothly on compact subsets of their limiting plane or half-plane. Here the details of [F2] apply without removing points: the global derivative bound two already gives local equicontinuity everywhere. Compact target finite nets and a countable dense subset of the limiting domain give a locally uniform subsequence. In a small disk about each limiting source point choose the fixed target coordinates of [F2], normalizing $J_{\ast }$ to $i$ at the center; at boundary points use its explicit totally real chart. Locally uniform convergence and smooth coefficient convergence then give the coefficient bound $\mathrm{\parallel }J-i\mathrm{\parallel }\le 1\mathrm{/}2$ and uniform coordinate $C^1$ bounds on a smaller fixed patch. The bootstrap in [F2] gives bounds of every order there. Its countable derivative diagonal construction yields local $C^{\mathrm{\infty }}$ convergence on an exhaustion of the entire limiting domain, including boundary derivatives. Thus no concentration point remains in this selected scale.

Let $v$ be the limit. The equation and boundary condition pass to the limit. Its derivative norm is one at zero in the plane-center case, and at $ih$ in the bounded-height case: convergence is $C^1$ on a fixed compact neighborhood of those moving points. Its derivative norm is at most two by the same convergence, so it is nonconstant. A translation and dilation of a two-dimensional source preserves Dirichlet energy, since the squared derivative scaling cancels the area scaling. Therefore every compact-region energy of the rescaled maps is at most $E_0$, and passage to the smooth limit followed by exhaustion gives $E(v)\le E_0$.

Finally choose fixed constants $r_{\ast }>0$, ${\epsilon }_{\ast }>0$, and $C_{\ast }>0$ giving the interior center and boundary half-radius supremum estimates of [F1] in the metric $g$, uniformly for every possible autonomous limit $J_{\ast }$. This is possible from the uniform coefficient bounds and the two fixed boundary structures; decrease $r_{\ast }$ below $1\mathrm{/}4$ if necessary. If a derivative-norm-one point of $v$ in the plane, or in the half-plane at height at least $r_{\ast }\mathrm{/}4$, is considered, apply the interior estimate on a radius-$r_{\ast }\mathrm{/}8$ disk about it. If its half-plane height is below $r_{\ast }\mathrm{/}4$, apply the boundary supremum estimate on the radius-$r_{\ast }$ half-disk centered at its boundary foot. If total energy is at least ${\epsilon }_{\ast }$ there is already a positive lower bound. Otherwise the relevant small-energy estimate gives $1\le \frac{64C_{\ast }}{r_{\ast }^2}E(v)\mathrm{.}$ Consequently $\mathrm{\hslash }=\min \{{\epsilon }_{\ast },r_{\ast }^2\mathrm{/}(64C_{\ast })\}>0$ is a uniform lower bound for the energy of this nonconstant bubble. This proves exactly the asserted plane or half-plane result, without filling its point at infinity.