Patrícia Kruse Klaser

arXiv2026

Invariant solutions for the asymptotic Plateau problem in $\mathbb{H}^3$

In this paper, we present solutions to the asymptotic Plateau problem in the hyperbolic space $\mathbb{H}^3$. In this context, we exhibit solutions for curves that are invariant under the action of a one-parameter subgroup of isometries of $\mathbb{H}^3$. To achieve this, we prove the existence of foliations of $\mathbb{H}^3$ by minimal surfaces that are properly embedded, complete, and invariant under these subgroups, which are then used to solve the problem.

math.DG↗

arXiv2015

Bounded $λ$-harmonic functions in domains of $\mathbb{H}^n$ with asymptotic boundary with fractional dimension

The existence and nonexistence of $λ$-harmonic functions in unbounded domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $Ω$ is zero, then there is no bounded $λ$-harmonic function of $Ω$ for $λ\in [0,λ_1(\mathbb{H}^n)]$, where $λ_1(\mathbb{H}^n)=(n-1)^2/4$. For these domains, we have comparison principle and some maximum principle. Conversely, for any $s>(n-1)/2,$ we prove the existence of domains with asymptotic boundary of dimension $s$ for which there are bounded $λ_1$-harmonic functions that decay exponentially at infinity.

math.AP↗