23.1.31 A base correction and unbounded mass

Adding a function of $x$ alone changes $Q[W]$ by its second derivative and leaves the vertical energy unchanged. The integral bound for $d_\eta $ lets us choose this second derivative larger than $d_\eta $ and compensate its integral on an interval where $Q[W]$ is uniformly positive. The correction has small first derivative, so the remaining horizontal coefficient stays positive as well.

of theorem 23.1.1. Choose a smooth function $r_\eta \geq 0$ supported in $(-1/2,b+1/2)$ with

\[ r_\eta >d_\eta \ \hbox{on }[0,b],\qquad m_\eta :=\int _\mathbb Rr_\eta \, \mathrm{d}x\leq \int _\mathbb Rd_\eta \, \mathrm{d}x+\eta =o(1). \]

Such a majorant exists by uniform approximation of the compactly supported continuous function $d_\eta $, followed by adding a positive bump on $[0,b]$; for each fixed $\eta $ the approximation error can be chosen small enough for the integral bound. Fix a smooth $\psi \geq 0$ supported in $(-2,-1)$ with integral one and set

\[ g_\eta (x)=\int _{-\infty }^ x(x-s) (r_\eta (s)-m_\eta \psi (s))\, \mathrm{d}s, \qquad V_\eta =W+g_\eta . \]

The negative part of $g_\eta ''$ lies to the left of its positive part, and their integrals agree. Consequently,

\[ -m_\eta \leq g_\eta '\leq 0, \qquad g_\eta '=0\ \hbox{for }x\leq -2 \ \hbox{and for }x\geq b+1/2. \]

Thus $g_\eta $ vanishes for $x\leq -2$ and is constant for $x\geq b+1/2$, and $V_\eta $ has the properties at the two ends required for smooth extension to $X$.

Since $g_\eta $ is independent of $w$, the Schur complement changes by exactly its second derivative:

\[ Q[V_\eta ]=Q[W]+r_\eta -m_\eta \psi . \]

This is positive on $[0,b]$ because $r_\eta >d_\eta $ there. On $(-2,-1)$ it is at least $2/27-m_\eta \| \psi \| _\infty >0$ for small $\eta $. Elsewhere $Q[W]>0$, $\psi =0$ and $r_\eta \geq 0$. The vertical coefficient remains above $1-\eta $ by (23.1.24). For the remaining coefficient, if $x\leq -2$, both $W_ x$ and $g_\eta '$ vanish; if $-2\leq x\leq 0$, then $W_ x=0$ and $\mu \geq \mu (-2)>0$, while $|g_\eta '|\leq m_\eta =o(1)$; and if $x\geq 0$, then $\mu \geq 1/2$ and $W_ x\geq -o(1)$. In all cases $\mu +(V_\eta )_ x>0$ for sufficiently small $\eta $. Therefore lemma 23.1.7 gives $\omega +\mathrm{dd}^{\mathrm{c}}V_\eta >0$ on $X$.

On the fixed ball $B=\{ |\xi |^2<\mathrm{e}^{-3}\} \Subset U_0$, one has $V_\eta =F_\eta (\varepsilon ,w)$, including at $\xi =0$. Thus

\[ \int _{\pi ^{-1}(y)}|(V_\eta )_ w|^2\mathrm{d}A_ E =2\eta ^2\log (1/\varepsilon )+O(\eta ^2) =2\log (1/\eta )+O(\eta ^2)\longrightarrow +\infty \]

uniformly for $y\in B$. Since $\int _ B\omega _{\mathrm{FS}}^2>0$, the mass identity (23.1.5) shows that $\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}V_\eta )^3\to +\infty $, and the theorem follows by taking $V_ j=V_{\eta _ j}$ along any sequence $\eta _ j\downarrow 0$. $\square$

23.1.31 A base correction and unbounded mass`D7D1053AF199`

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23.1.31 A base correction and unbounded massD7D1053AF199

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23.1.31 A base correction and unbounded mass`D7D1053AF199`