Positive energy gap for nonconstant curves on a fixed closed domain

If the target is empty or zero-dimensional, no nonconstant map from the connected source exists and any positive gap suffices. Otherwise cover the fixed compact source by finitely many conformal coordinate disks and smaller concentric disks, using a radius no larger than the $r_0$ in [F1]. On their compact closures the fixed conformal metric factors have positive finite upper and lower bounds. Energy is conformally invariant in two source dimensions. Consequently there are $\epsilon >0$ and $C>0$ such that $E_g(u)<\epsilon$ implies the global estimate ${\sup }_{\mathrm{\Sigma }}\mathrm{\mid }du{\mathrm{\mid }}_{g,h}\le C\sqrt{E_g(u)}$: apply [F1] at each point on a disk of a fixed smaller radius in one of the finitely many charts, and use the finite metric comparison constants.

In target coordinates the autonomous equation is $u_x+J(u)u_y=0$. Differentiating in $x$ and $y$ and using $J^2=-I$ gives $u_{xx}+u_{yy}=-(dJ(u)u_x)u_y+J(u)(dJ(u)u_y)u_y\mathrm{.}$ Thus on a relatively compact target chart $\mathrm{\mid }\mathrm{\Delta }u\mathrm{\mid }\le C_0\mathrm{\mid }du{\mathrm{\mid }}^2$ with a uniform coefficient constant. For a coordinate center $a$ the function $f=\mathrm{\mid }u-a{\mathrm{\mid }}^2$ therefore satisfies $\mathrm{\Delta }f\ge (2-2C_0\mathrm{\mid }u-a\mathrm{\mid })\mathrm{\mid }du{\mathrm{\mid }}^2$. Choose a smaller coordinate ball so $2C_0\mathrm{\mid }u-a\mathrm{\mid }\le 1$. A finite cover of compact $X$, with smaller chart neighborhoods inside the original ones, gives one target metric radius $\delta >0$ such that every metric ball $B_g(p,\delta )$ is contained in such a coordinate ball centered at the coordinates of $p$, with this inequality. If a closed-source map has image in that ball, integrate the inequality against its conformal source metric: the integral of its Laplacian is zero on the closed surface, by integration by parts in finitely many charts with a finite smooth partition of unity (the partition-derivative terms cancel), whereas the nonnegative right side vanishes only if $du=0$. Such a map is constant.

Let $D$ be a finite upper bound for lengths of source paths joining any two points of the connected compact Riemannian surface. To construct such a bound without a completeness theorem, choose finitely many smaller coordinate balls covering the source and larger relatively compact coordinate balls containing their closures. Their intersection graph is connected, since otherwise their two disjoint unions would disconnect the source. Choose one joining point for each edge of a finite spanning tree. Straight coordinate segments within a larger coordinate ball have a uniform finite length bound; concatenating them along the finite tree joins arbitrary source points with one uniform bound $D$. If $E_g(u)<\epsilon$, step 1.1 bounds the diameter of its image by $CD\sqrt{E_g(u)}$. Choose $\mathrm{\hslash }=\min \{\frac{\epsilon }{2},\frac{{\delta }^2}{4C^2\max \{D,1{\}}^2}\}>0.$ Energy less than $\mathrm{\hslash }$ puts the image inside $B_g(u(z_0),\delta )$ for any fixed $z_0$, so step 1.2 makes the map constant. The contrapositive proves the claimed gap. This is a fixed-domain gap and is not asserted uniform over degenerating source structures.