Related papers: Polyharmonic hypersurfaces into complex space form…
We find the necessary and sufficient condition for the proper biharmonic CMC hypersurfaces in the special warped product space. Furthermore, we obtain that there exists no proper biharmonic CMC compact hypersurface there.
These notes are concerned with harmonic and holomorphic functions on Euclidean spaces, using quaternions and Clifford algebras in higher dimensions. The main themes are weak solutions, the mean-value property, and subharmonicity.
In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit…
We present results expressing conditions for the existence of meromorphic first integrals for Pfaff equations of arbitrary codimension, integrable or not, on complex manifolds. These results are in the same vein as previous ones by J-P.…
We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work…
We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.
We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.
We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$…
Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one…
We prove that the rational cohomology of the space of non-singular complex homogeneous polynomials of degree d in a fixed number of variables stabilizes to the cohomology of the general linear group for d sufficiently large.
In this paper, we study \lambda-biharmonic hypersurfaces in the product space L^{m}\times\mathbb{R}, where L^{m} is an Einstein space and \mathbb{R} is a real line. We prove that \lambda-biharmonic hypersurfaces with constant mean curvature…
Non-polynomial growth harmonic maps from the complex plane to the hyperbolic space are studied. Some non-surjectivity results are obtained. Moreover, images of such harmonic maps are investigated with reference to their Hopf differentials.
A Hopf hypersurface in complex hyperbolic space CH^n is one in which the complex structure applied to the normal vector is a principal direction at each point. In this paper, Hopf hypersurfaces for which the corresponding principal…
We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular,…
It is known that the $L^{2}$-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the $L^{2}$-norms of harmonic functions over a wide class of evolving…
We develop new elements of harmonic analysis on the complex sphere on the basis of which Bernstein's, Jackson's and Kolmogorov's inequalities are established. We apply these results to get order sharp estimates of $m$-term approximations.…
The main aim of this paper is to prove the existence of certain proper weakly $r$-harmonic ($ES-r$-harmonic) maps. We construct critical points which belong to a family of rotationally symmetric maps $\varphi_a : B^n \to \mathbb{S}^n$,…
We provide normal forms for singularities of analytic hypersurfaces in $({\mathbb C}^n,0)$, using holomorphic vector fields.
In this paper we construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb C^n$ (i.e., a long $\mathbb C^n$), but it does not admit any nonconstant…
Harmonic maps are nonlinear extensions of harmonic functions. They are critical points of natural energy functionals between Riemannian manifolds. Such type of problems appear in Physics, Geometry of Finance and the study of regularity and…