Higgs bundles and applications

This short note gives an overview of how a few conjectures and theorems of the author and collaborators fit together. It was prepared for Oberwolfach's workshop Differentialgeometrie im Gro{\ss}en, 28 June - 4 July 2015, and contains no new results.Comment: 3 page

On equivalence theorems of Minkowski spaces and applications in Finsler geometry

In this paper, we first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As an application in Finsler geometry, we gives some new characterizations of Berwald spaces.Comment: In this new version, we modify some parts of the frist versio

A Characterisation of Manhart's Relative Normal Vector Fields

In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" which generalises the notion of unit normal vector field N from Euclidean differential geometry. A concise review of relative differential geometry will be presented. The main result, to which the title of this article refers, will be given in the third section. Here we consider, for a function $f$ of two variables, relative normal vector fields of the form f(H,K)\,N-\grad_{\II}(f(H,K)) for non-degenerate surfaces in the Euclidean three-dimensional space. A comparison of the variation of the curvature functional \int f(H,K)\,\dd\Omega with the relative area functional obtained from the above relative normal vector field, results in a distinguishing property for the one-parameter family of relative normal vector fields which was introduced by F.\ Manhart, and which is obtained by choosing $f(H,K)=|K|^{\alpha}$ (where we will assume that $\alpha\neq 1$). More precisely, the following will be shown in theorem~6: ``{The curvature functionals $(\ast)$ for which the critical points coincide with the relative-minimal surfaces with respect to the relative normal vector field $(\dagger)$, are essentially those obtained from Manhart's family}." In the fourth section, we give a characterisation of the sphere by means of relations between the support function and the curvatures. In the last section, we combine the previously described results and arrive at a variational characterisation of the sphere.Comment: 12 pages, 1 figur

A relative-geometric treatment of ruled surfaces

We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are characterized by the support functions $^{\left( \alpha \right) }q=\left \vert K\right \vert ^{\alpha}$ for $\alpha \in \mathbb{R}$ (Manhart's relative normalizations). All ruled surfaces for which the relative normals, the Pick invariant or the Tchebychev vector field have some specific properties are determined. We conclude the paper by the study of the affine normal image of a non-conoidal ruled surface.Comment: 12 page

Differential geometry of general affine plane curves

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature and signature are the complete invariants of regular curves. As application we give a complete classification of constant curvature curves in $A^2$

On the Tchebychev Vector Field in the Relative Differential Geometry

In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space $\mathbb{R}^{n+1}$. Considering a relative normalization $\bar{y}$ of an hypersurface $\Phi$ we decompose the corresponding Tchebychev vector $\bar{T}$ in two components, one parallel to the Tchebychev vector $\bar{T}_{EUK}$ of the Euclidean normalization $\bar{\xi}$ and one parallel to the orthogonal projection $\bar{y}_{T}$ of $\bar{y}$ in the tangent hyperplane of $\Phi$. We use this decomposition to investigate some properties of $\Phi$, which concern its Gaussian curvature, the support function, the Tchebychev vector field etc.Comment: 10 page

Quasilinear systems with linearizable characteristic webs

We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order constraints for the characteristic speeds of the system. It is demonstrated that every such system with n>3 components can be transformed by a reciprocal transformation to n uncoupled Hopf equations. All our considerations are local

On the Area Functional of the Second Fundamental Form of Ovaloids

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints.Comment: 13 pages. (v2: As compared to a previous version of this article, which was entitled "The Mean Curvature of the Second Fundamental Form of an Ovaloid", there have been made substantial changes especially to the latter part of the article.) (v3: Reference to the work of M. Wiehe has been added.) Published version available here: http://www.emis.de/journals/MPRIA/100152.htm

A note on two-dimensional minimal surface graphs in R^n and a theorem of Bernstein-Liouville type

Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouville type

Harmonic morphisms between degenerate semi-Riemannian manifolds

In this paper we generalize harmonic maps and morphisms to the \emph{degenerate semi-Riemannian category}, in the case when the manifolds $M$ and $N$ are \emph{stationary} and the map $\phi :M\to N$ is \emph{radical-preserving}. We characterize geometrically the notion of \emph{(generalized) horizontal (weak) conformality} and we obtain a characterization for (generalized) harmonic morphisms in terms of (generalized) harmonic maps.Comment: 21 pages, LaTe