Liouville theorem, conformally invariant cones and umbilical surfaces for Grushin-type metrics

进入共形平坦空间的 $f$-双调和超曲面

$f$-Biharmonic hypersurfaces into a conformally flat space

本文研究共形平坦及非正曲率流形中的 $f$-双调和超曲面，证明全脐情形的非紧性与极小性，并构造、分类相关曲面及高维真 $f$-双调和子流形。

Totally umbilical surfaces in three-manifolds with a parallel null vector field

We study non-degenerate, totally umbilical surfaces of a special class of pseudo-Riemannian manifolds, namely Walker three-manifolds. We show that such surfaces are either one of a totally geodesic family described by Calvaruso and Van der Veken or the ambient manifold must be locally conformally flat (here the surface can also be totally geodesic). The proof makes use of a key technique deployed by Manzano and Soaum in their recent classification of totally umbilical surfaces in homogeneous Riemannian three-manifolds.

Spherical 2-Dupin submanifolds

We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.

Umbilical-Type Surfaces in Spacetime

A spacelike surface S immersed in a 4-dimensional Lorentzian manifold will be said to be umbilical along a direction N normal to S if the second fundamental form along N is proportional to the first fundamental form of S. In particular, S is pseudo-umbilical if it is umbilical along the mean curvature vector field H, and (totally) umbilical if it is umbilical along all possible normal directions. The possibility that the surface be umbilical along the unique normal direction orthogonal to H --- "ortho-umbilical" surface--- is also considered. I prove that the necessary and sufficient condition…

Umbilical submanifolds of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$

In this article we complete the classification of the umbilical submanifolds of a Riemannian product of space forms, addressing the case of a conformally flat product $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$, which has not been covered in previous works on the subject. We show that there exists precisely a $p$-parameter family of congruence classes of umbilical submanifolds of $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$ with substantial codimension~$p$, which we prove to be at most $\mbox{min}\,\{k+1, n-k+2\}$. We study more carefully the cases of codimensions one and two and exhibit, respectively,…

关于双调和共形超曲面

On biharmonic conformal hypersurfaces

本文研究了共形超曲面的双调和性质，证明了在特定条件下共形因子必须为等参数函数，并展示了某些超曲面在特定空间形式中的双调和浸入性。