Blaschke Dupin hypersurfaces and equiaffine tubes

We give constructions of Blaschke Dupin hypersurfaces and a Blaschke isoparametric ones in terms of the notion of an equiaffine tube. In particular, the construction of Blaschke isoparametric hypersurfaces includes the Calabi-type composition of improper affine spheres (or an improper one and a proper one).     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

The norm theorem for semisingular quadratic forms

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of $F[x_1,\dots ,x_n]$ becomes a norm of such a quadratic form over the rational function field $F(x_1,\dots ,x_n)$. This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2.     

Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2n‐dimensional nondegenerate hypersurfaces ruled by n‐planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.     

A Bernstein theorem for affine maximal-type hypersurfaces

We obtain, in any dimension N and for a large range of values of θ, a Bernstein theorem for the fourth-order partial differential equation of affine maximal type

$u^{ij}{D}_{ij}w=0,w={[\det ?D^{2}u]}^{?\theta }$

assuming the completeness of Calabi's metric. This contains the results of Li–Jia [A.M. Li, F. Jia, Ann. Glob. Anal. Geom. 23 (2003)] for affine maximal equations and of Zhou [B. Zhou, Calc. Var. Partial Differ. Equ. 43 (2012)] for Abreu's equation. In particular, we extend the result of Zhou from $2\le N\le 4$ to $2\le N\le 5$.     

Stability properties and gap theorem for complete f-minimal hypersurfaces

In this paper, we study complete oriented f -minimal hypersurfaces properly immersed in a cylinder shrinking soliton \((\mathbb{S}^n \times \mathbb{R},\bar g,f)\).We prove that such hypersurface with L _f -index one must be either \(\mathbb{S}^n \times \{ 0\}\) or \(\mathbb{S}^{n - 1} \times \mathbb{R}\), where \({S}^{n - 1}\) denotes the sphere in \(\mathbb{S}^n\) of the same radius. Also we prove a pinching theorem for them.     

The Weierstrass factorization theorem for slice regular functions over the quaternions

The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains natural quaternionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular functions. This result holds in a formulation that reflects the peculiarities of the quaternionic setting and the structure of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions.     

The second main theorem of hypersurfaces in the projective space

We deal with a holomorphic map from the complex plane ${\mathbb{C}}$ to the n-dimensional complex projective space ${\mathbb{P}^{n}(\mathbb{C})}$ and prove the Nevanlinna Second Main Theorem for some families of non-linear hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ . This Second Main Theorem implies the defect relation. If the degree of the hypersurfaces are sufficiently large, the defect of the map is smaller than one. This means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate. To prove the Second Main Theorem, we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces. A meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the Wronskian operator. By resolving the singularity and pulling back a meromorphic partial projective connection, we also prove the Second Main Theorem for singular hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ , and prove the Second Main Theorem for smooth hypersurfaces in ${\mathbb{P}^{2}(\mathbb{C})}$ which are not normal crossing.     

A sewing theorem for quadratic differentials

Quadratic differentials \mathfrakQ(z)dz² \mathfrak{Q}(z)d{z^2} on a finite Riemann surface with poles of order not exceeding two are considered. The existence of such a differential with prescribed metric characteristics is proved. These characteristics are the following: the leading coefficients in the expansions of the function \mathfrakQ(z) \mathfrak{Q}(z) in neighborhoods of its poles of order two, the conformal modules of the ring domains, and the heights of the strip domains in the decomposition of the Riemann surface defined by the structure of trajectories of this differential. Bibliography: 5 titles.     

On the Teichmüller theorem and the heights theorem for quadratic differentials

By using the Marden-Strebel heights theorem for quadratic differentials, we provide a concrete method for finding the Teichmüller differential associated with the Teichmüller mapping between compact or finitely punctured Riemann surfaces.

     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.