On the biharmonic and harmonic indices of the Hopf map

Exemples de sous variétés de $\mathbb{S}^n$; les variétés proprement biharmoniques avec indice et les variétés proprement C--harmoniques

We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biharmonic (i.e. proper indexed-biharmonic). We give some pseudo-umbilical and non pseudo-umbilical examples and we study the link between the proper indexed-biharmonic submanifolds of $\mathbb{S}^n$ and the proper conformal-harmonic submanifolds of $\mathbb{S}^n$.

Survey on the Biharmonic Hypersurfaces in Terms of the Induced Metric of Tensor Ricci

In this article, we study the biharmonic hypersurfaces in the Sasakian space form with the induced metric of tensor Ricci. We find the existence necessary and sufficient condition of the biharmonic hypersurfaces there. We show that the biharmonic Hopf hypersurfaces are minimal where gradient of the mean curvature is in direction of the structural vector fields. Furthermore, we prove that does not exist any biharmonic Hopf hypersurfaces when the gradient of the mean curvature is a principal direction.

Explicit formulas for biharmonic submanifolds in non-Euclidean 3-spheres

We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.

指标与留数：从庞加莱-霍普夫（Poincaré-Hopf）到鲍姆-博特（Baum-Bott），以及马尔科·布鲁内拉（Marco Brunella）

Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella

该综述性文章研究向量场与全纯叶状结构的不变量，旨在关联复解析奇异簇与复流形上奇异全纯叶状结构这两个有诸多共通点的理论领域。

k-Harmonic immersion and submersion into a sphere

J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, We study k-harmonic immersion into a sphere, and get the rerationship between radious and "k" of k-harmonic. And we also consider k-harmonic submersion, and genelarize Oniciuc's results. Futhermore we construct non harmonic k-harmonic by Hopf map.

Energy identity for approximations of harmonic maps from surfaces

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.