Boundary behaviors for Liouville's equation in planar singular domains

Abstract

Abstract. We study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary. The conformal structure plays an essential role.

Minimal graphs in the hyperbolic space with singular asymptotic boundaries

“…This is partly due to the diversity of singularity and complexity of the relevant geometric problems. The first two authors studied the asymptotic behaviors of solutions of the Liouville equation in [5] and solutions of the Loewner-Nirenberg problem in [6] in singular domains and proved that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary.…”

A boundary expansion of solutions to nonlinear singular elliptic equations

“…where a j is a smooth function, ϕ 0 is a defining function of the domain, the function u m ∈ C m (Ω), and m is an integer greater than N . Similar expressions were given in [1,11] for the Loewner-Nirenberg problem, in [5,7] for the analysis of the boundary conformal structure, and in [9] for equations from a conformal anomaly of submanifold observables. In [10], a similar formula was proved for minimal graphs in the hyperbolic space.…”

Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities

“…Consider the case that near a boundary point, say the origin, Ω coincides with the first quadrant in R 2 near the origin. We check that u T = log( 1 2 r sin(2θ)) = log( 1 [11], and satisfies…”

“…The case when ∂Ω is singular was studied by del Pino and Letelier [7], Marcus and Veron [18], and Han and Shen [12]. See also Han and Shen [11] for the Liouville's equation in planar singular domains.…”