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Patrícia Klaser

Publications and source records attributed to Patrícia Klaser.

4 recordsLinked to original sources
arXiv2020

On the asymptotic Plateau problem for area minimizing surfaces in $\mathbb{E}(-1,τ)$

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,τ)$. As one of our main results, we present sufficient conditions for a curve $Γ$ in $\partial_{\infty} \mathbb{E}(-1,τ)$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,τ)$ having $Γ$ as its asymptotic boundary.

math.DG↗
arXiv2020

CMC Graphs With Planar Boundary in $\mathbb{H}^{2}\times \mathbb{R}$

It is known that for $Ω\subset \mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $G\subset \mathbb{R}^{3}$ of constant mean curvature $H$ over $Ω$ with $\partial G=$ $\partial Ω$ if and only if $Ω$ is included in a strip of width $1/H$. In this paper we obtain results in $\mathbb{H}^{2}\times \mathbb{R}$ in the same direction: given $H\in \left( 0,1/2\right) $, if $Ω$ is included in a region of $\mathbb{ H}^{2}\times \left\{ 0\right\} $ bounded by two equidistant hypercycles $\ell(H)$ apart, we show that, if the geodesic curvature of $\partial Ω$ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Ω$ with $\partial G=\partial Ω$. We also present more refined existence results involving the curvature of $\partialΩ,$ which can also be less than $-1.$

math.DG↗
arXiv2014

Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

In a Hadamard manifold $M$, it is proved that if $u$ is a $λ$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $λ$ and on the dimension of $M$. This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p,λ$ and $H.$

math.DG↗