Pome

Theorem 10.22.7FF884313C070

Theorem 10.22.7.FF884313C070 For the normalized family in theorem 10.22.1, put $L=\log (1/s)$ for $0<s<1$.

  1. If $D(s)\sim Cs^{\ell -n+1}L^{-b}$ with $C>0$, $\ell >-1$ and $b\in \mathbb R$, put $\gamma =(\ell +1)/n$ and $K=(C/\gamma )^{1/n}$. Then

    940248BAAEE2 10.22.8
    \begin{equation} m\sim Ks^\gamma L^{-b/n},\qquad U(s)-U(0)\sim \frac K\gamma s^\gamma L^{-b/n},\qquad m'\sim \gamma K s^{\gamma -1}L^{-b/n}. \end{equation}

    The residual is bounded and Hölder continuous. In bounded blowup charts, its metric is comparable to $L^{-b/n}(|z|^{2\gamma -2}|\mathrm{d} z|^2+|z|^{2\gamma }|\mathrm{d} w|^2)$, and its distance to $E$ is of order $|z|^\gamma (\log (1/|z|))^{-b/(2n)}$. In particular, the metric is incomplete near $E$.

  2. If $D(s)\sim Cs^{-n}L^{-b}$, the local mass is finite precisely for $b>1$. In that range put $a=(b-1)/n$ and $K=(nC/(b-1))^{1/n}$. Then

    CC16B5AD8D47 10.22.9
    \begin{equation} m\sim KL^{-a},\qquad m'\sim Ka s^{-1}L^{-a-1}. \end{equation}

    The potential has a finite limit at the origin, the solution has minimal singularities, and the distance is finite, each precisely when $b>n+1$; in that case $U(s)-U(0)\sim K L^{1-a}/(a-1)$. At $b=n+1$, $U$ diverges as $-K\log L$, and for $1<b<n+1$ as $-K L^{1-a}/(1-a)$. In these unbounded cases the solutions have full mass but not minimal singularities, and their metrics are complete near $E$. Their densities on the blowup lie in no $L^ p$ with $p>1$.

  3. If $D(s)\sim Cs^{-n}L^{-(n+1)}(\log L)^{-d}$ with $d\in \mathbb R$, put $C_*=C^{1/n}$. Then

    5C6F9700C172 10.22.10
    \begin{equation} m\sim C_*L^{-1}(\log L)^{-d/n},\qquad m'\sim C_*s^{-1}L^{-2}(\log L)^{-d/n}. \end{equation}

    The potential is bounded precisely when $d>n$, with $U(s)-U(0)$ of order $(\log L)^{1-d/n}$. At $d=n$ it diverges as $-C_*\log \log L$, and for $d<n$ as $-C_*(\log L)^{1-d/n}/(1-d/n)$. The distance is finite precisely when $d>2n$, with remaining length of order $(\log L)^{1-d/(2n)}$. Thus for $n<d\leq 2n$ the residual is bounded and continuous but not Hölder, the solution has minimal singularities, and the metric is complete near $E$. All these critical densities have finite mass and lie in no $L^ p$ with $p>1$.

In all cases the logarithmic coefficient of the solution along $E$ is $c$, whereas by proposition 15.23.2 a strict barrier can be chosen with coefficient $c+\delta $, for some $\delta >0$ independent of $\ell ,b,d$.

Proof. We use the exact expressions (10.22.3). Each asymptotic formula follows by bounding $D$ between $(1\pm \epsilon )$ times its model and integrating the model; no derivative of an unspecified error term is taken. In the first case $m^ n\sim nC s^{\ell +1}L^{-b}/(\ell +1)$, which yields (10.22.8). Both Hessian eigenvalues are then comparable to $s^{\gamma -1}L^{-b/n}$, and the substitution of blowup coordinates gives the assertions about the metric and the distance. The positive power of $s$ makes the potential Hölder continuous, after the logarithmic factor is absorbed by a slightly smaller exponent.

In the second case, $n\int _0^ s t^{-1}C(\log (1/t))^{-b}\, \mathrm{d} t$ is finite precisely for $b>1$, and it is then asymptotic to $nC L^{1-b}/(b-1)$. This gives (10.22.9). Since $\mathrm{d} U=-m\, \mathrm{d} L$, boundedness of $U$ is equivalent to convergence of $\int ^\infty L^{-a}\, \mathrm{d} L$, that is, to $a>1$. The distance is a constant multiple of $\int ^\infty L^{-(a+1)/2}\, \mathrm{d} L$ and has the same strict threshold. Explicit primitives give the stated finite and divergent behavior, including the logarithm at $a=1$. On the blowup the density is comparable to $|z|^{-2}(\log (1/|z|))^{-b}$; for every $p>1$, its $p$th power acquires an exponentially growing factor in logarithmic coordinates, so the density is not in $L^ p$.

In the third case, $m^ n\sim C L^{-n}(\log L)^{-d}$. The criteria for the potential and for the distance become convergence of

\[ \int ^\infty L^{-1}(\log L)^{-d/n}\, \mathrm{d} L \quad \text{and}\quad \int ^\infty L^{-1}(\log L)^{-d/(2n)}\, \mathrm{d} L, \]

respectively. Their thresholds are $d>n$ and $d>2n$, and direct integration gives the stated tails and divergent behavior. A bounded tail of order a positive power of $\log L$ tends to zero more slowly than every positive power of $|z|$, so the potential is not Hölder continuous. The mass integral is finite for every real $d$ because the principal factor is $L^{-(n+1)}$ with $n\geq 2$, and the same exponential argument excludes $L^ p$ for $p>1$. In particular, all the endpoint cases occur within the prescribed family of theorem 10.22.1. $\square$

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