2014 / June

Singularities for A Fully Non-Linear Elliptic Equation In Conformal Geometry

Maria del Mar Gonzalez, Lorenzo Mazzieri

$\delta_k$-curvature, fully nonlinear elliptic equations, conformal geometry, singular metrics, $\delta_k$-curvature, fully nonlinear elliptic equations, conformal geometry, singular metrics

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223-244

We construct some radially symmetric solutions of the constant $\sigma_k$-equation on $\mathbb R^n \setminus \mathbb{R}^p$, which blow up exactly at the submanifold $\mathbb R^p \subset \mathbb R^n$. These are the basic models to the problem of finding complete metrics of constant $\sigma_k$--curvature on a general subdomain of the sphere $\mathbb S^n\backslash\Lambda^p$ that blow up exactly at the singular set $\Lambda^p$ and that are conformal to the canonical metric. More precisely, we look at the case $k=2$ and $ 0 < p < p_2 := \frac{n-\sqrt n-2}{2} $. The main result is the understanding of the precise asymptotics of our solutions near the singularity and their decay away from the singularity. The first aspect will insure the completeness of the metric about the singular locus, whereas the second aspect will guarantee that the model solutions can be locally transplanted to the original metric on $\mathbb S^n$, and hence they can be used to deal with the general problem on $\mathbb S^n\backslash\Lambda^p$.

35J75, 53C21, 58J60

2013-08-04

2014-03-18