arXiv subjects

Pedro Fusieger

Publications and source records attributed to Pedro Fusieger.

3 recordsLinked to original sources

arXiv2019

Asymptotic and exterior Dirichlet problems for the minimal surface equation in the Heisenberg group with a balanced metric

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}% \nolimits_{3}.$ It is then proved the existence of complete properly embedded minimal surfaces in $\operatorname*{Nil}\nolimits_{3}$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. It is also proved the existence of complete properly embedded minimal surfaces foliating an open set of $\operatorname*{Nil}\nolimits_{3}$ having as boundary a given curve $Γ$ in $\mathbb{T},$ satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of $\mathbb{T}\backslashΓ$.

math.DG↗

arXiv2019

Gauss map and the topology of constant mean curvature hypersurfaces of $\mathbb{S}^{7}$ and $\mathbb{CP}^{3}$

We define a Gauss map $γ:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $γ$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $γ$, are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of $\mathbb{CP}^{3}$ and obtain similar results for CMC hypersurfaces of this space.

math.DG↗

arXiv2018

Minimal isoparametric submanifolds of $\mathbb{S}^{7}$ and octonionic eigenmaps

We use the octonionic multiplication $\cdot$ of $\mathbb{S}^{7}$ to associate, to each unit normal section $η$ of a submanifold $M$ of $\mathbb{S}^{7},$ an octonionic Gauss map $γ_η:M\rightarrow\mathbb{S}^{6},$ $γ_η(x)=x^{-1}\cdotη(x),$ $x\in M,$ where $\mathbb{S}^{6}$ is the unit sphere of $T_{1}\mathbb{S}^{7},$ $1$ is the neutral element of $\cdot$ in $\mathbb{S}^{7}.$ Denoting by $\mathcal{N}(M)$ the vector bundle of normal sections of $M$ we set, for $η$ $\in\mathcal{N}(M),$ $S_η(X)=-\left(\nabla_{X}η\right) ^{\top},$ $X\in TM.$ Considering the Hilbert-Schmidt inner product on the vector bundle $\mathcal{S}(M)=\left\{S_η, \ \text{}η\in\mathcal{N}(M)\right\} $ and defining the bundle map $\mathcal{B} :\mathcal{N}(M)\rightarrow\mathcal{S}(M)$ by $\mathcal{B}(η)=S_η,$ we prove that if $M$ is a minimal submanifold of $\mathbb{S}^{7}$ and $η\in\mathcal{N}(M)$ is unitary and parallel on the normal connection, then $γ_η$ is harmonic if and only if $η$ is an eigenvector of $\mathcal{B}^{\ast}\mathcal{B}:\mathcal{N}(M)\rightarrow\mathcal{N}(M),$ where $\mathcal{B}^{\ast}$ is the adjoint of $\mathcal{B}.$ If $M$ is an isoparametric compact minimal submanifold of codimension $k$ of $\mathbb{S}% ^{7}$ then $\mathcal{B}^{\ast}\mathcal{B}$ has constant non negative eigenvalues $0\leqσ_{1}\leq\cdots\leqσ_{k}$ and the associated eigenvectors $η_{1},\cdots,η_{k}$ form an orthonormal basis of $\mathcal{N}(M)$, parallel on the normal connection, such that each $γ_{η_{j}}$ is an eigenmap of $M$ with eigenvalue $7-k+$ $σ_{j}.$ Moreover, $σ_{j}=\Vert S_{η_{j}}\Vert^{2},$ $1\leq j\leq k.$

math.DG↗