A note on surfaces with parallel mean curvature
We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
math.DG↗
arXiv subjects
Dorel Fetcu
Publications and source records attributed to Dorel Fetcu.
A Simons type formula for surfaces with parallel mean curvature
We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
math.DG↗
Surfaces with parallel mean curvature vector in complex space forms
We consider a quadratic form defined on the surfaces with parallel mean curvature vector of an any dimensional complex space form and prove that its $(2,0)$-part is holomorphic. When the complex dimension of the ambient space is equal to $2$ we define a second quadratic form with the same property and then determine those surfaces with parallel mean curvature vector on which the $(2,0)$-parts of both of them vanish. We also provide a reduction of codimension theorem and prove a non-existence result for $2$-spheres with parallel mean curvature vector.
math.DG↗
Surfaces with parallel mean curvature in $\mathbb{C}P^n\times\mathbb{R}$ and $\mathbb{C}H^n\times\mathbb{R}$
We consider surfaces with parallel mean curvature vector (pmc surfaces) in $\mathbb{C}P^n\times\mathbb{R}$ and $\mathbb{C}H^n\times\mathbb{R}$, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant pmc $2$-spheres of a $5$-dimensional non-flat cosymplectic space form of product type are actually the embedded rotational spheres $S_H^2\subset\bar M^2\times\mathbb{R}$ of Hsiang and Pedrosa, where $\bar M^2$ is a complete simply-connected surface with constant curvature. When the ambient space is a cosymplectic space form of product type and its dimension is greater than $5$, we prove that an immersed non-minimal non-pseudo-umbilical anti-invariant $2$-sphere lies in a product space $\bar M^4\times\mathbb{R}$, where $\bar M^4$ is a space form. We also provide a reduction of codimension theorem for the pmc surfaces of a non-flat cosymplectic space form.
math.DG↗
A note on biharmonic curves in Sasakian space forms
We classify the biharmonic non-Legendre curves in a Sasakian space form for which the angle between the tangent vector field and the characteristic vector field is constant and obtain explicit examples of such curves in $\mathbb{R}^{2n+1}(-3)$.
math.DG↗