arXiv subjects

Julia Plehnert

Publications and source records attributed to Julia Plehnert.

3 recordsLinked to original sources

arXiv2015

Compact embedded minimal surfaces in $\mathbb{S}^2\times \mathbb{S}^1$

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$ via conjugate constructions. The resulting surfaces can be seen as the analogy to the Schwarz P-surface in these homogeneous 3-manifolds.

math.DG↗

arXiv2013

Constant mean curvature $k$-noids in homogeneous manifolds

For each $k\geq2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in[0,1/2]$ in $Σ(κ)\times\R$ where $κ+4H^2\leq0$. The surfaces are invariant under $2π/k$-rotations about a vertical fiber of $Σ(κ)\times\R$, have genus zero, and a finite number of ends. The first family generalizes the notion of $k$-noids: It has $k$ ends, one horizontal and $k$ vertical symmetry planes. The second family is less symmetric and has two types of ends. Each surface arises as the conjugate (sister) surface of a minimal graph in a homogeneous 3-manifold. The domain of the graph is non-convex in the second family. For $κ=-1$ the surfaces with constant mean curvature $H$ arise from a minimal surface in $\widetilde{\PSL}_2(\R)$ for $H\in(0,1/2)$ and in $\Nil$ for H=1/2. For H=0, the conjugate surfaces are both minimal in a product space.

math.DG↗

arXiv2013

Surfaces with constant mean curvature 1/2 and genus one in H2xR

We construct new constant mean curvature surfaces in H2xR. They arise as sister surfaces of Plateau solutions. It is a family of MC 1/2 surfaces with k ends, genus 1 and k-fold dihedral symmetry, k greater 2. The surfaces are Alexandrov- embedded.

math.DG↗