arXiv subjects

Hiba Bibi

Publications and source records attributed to Hiba Bibi.

4 recordsLinked to original sources

arXiv2024

Biharmonic Hypersurfaces in Euclidean Spaces

An isometric immersion $X: \Sigma^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic if $\Delta^2 X = 0$, i.e. if $\Delta H =0$, where $\Delta$ and $H$ are the metric Laplacian and the mean curvature vector field of $\Sigma^n$ respectively. More generally, biconservative hypersurfaces (BCH) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. We study and construct BCH that are holonomic, i.e. for which the principal curvature directions define an integrable net, and we deduce that $\Sigma^n$ is a holonomic biharmonic hypersurface iff it is minimal.

math.DG↗

arXiv2023

Segre embedding and biharmonicity

We consider the Segre embedding of the product $\mathbb{C}P^p\times\mathbb{C}P^q$ into $\mathbb{C}P^{p+q+pq}$ and study the biharmonicity of $M^p\times\mathbb{C}P^q$ and $M^p_1\times M^q_2$ as submanifolds of $\mathbb{C}P^{p+q+pq}$, where $M$ and $M_1$ are Lagrangian submanifolds of $\mathbb{C}P^p$ and $M_2$ is a Lagrangian submanifold of $\mathbb{C}P^q$. We find two new large classes of biharmonic submanifolds in complex projective space forms.

math.DG↗

arXiv2021

PMC Biconservative Surfaces in Complex Space Forms

In this article we consider PMC surfaces in complex space forms, and we study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in non-flat complex space forms and we prove that they are biconservative if and only if totally real. Then, we find a Simons type formula for a well-chosen vector field constructed from the mean curvature vector field. Next, we prove a rigidity result for CMC biconservative surfaces in 2-dimensional complex space forms. We prove then a reduction codimension result for PMC biconservative surfaces in non-flat complex space forms. We conclude by constructing from the Segre embedding examples of CMC non-PMC biconservative submanifolds, and we also discuss when they are proper-biharmonic.

math.DG↗

arXiv2020

Unique Continuation Property for Biharmonic Hypersurfaces in Spheres

We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.

math.DG↗