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We determine a primitive form for a universal unfolding of an affine cusp polynomial. Moreover, we prove that the resulting Frobenius manifold is isomorphic to the one constructed from the Gromov-Witten theory for an orbifold projective…
We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete…
A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper we show that the isomorphism class of the…
A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper $\delta$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric,…
Let $\bf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let $\Gamma=\bf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In…
The CSM class of a very affine manifold $U$ is represented by the rank drop locus of a general tuple of torus invariant 1-forms on it. This equality holds in the homology of any toric compactification $X\supset U$. It was proved for sch\"on…
We investigate the deformation theory of a class of generalized calibrations in Riemannian manifolds for which the tangent bundle has reduced structure group U(n), SU(n), G_2 and Spin(7). For this we use the property of the associated…
Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar\'e group $\pi_1(\pr^2 \se C)$ contains a free group with…
In this paper, we construct Poincar\'e-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal…
The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the…
For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional disk $\mathbb{D}^{2}$ satisfying certain uniqueness properties, we consider convex polygons $P\subset \mathbb{D}^{2}$ and define a metric $d$ on $P$ such that…
A principal torus bundle over a complex manifold with even dimensional fiber and characteristic class of type $(1,1)$ admits a family of regular generalized complex structures (GCS) with the fibers as leaves of the associated symplectic…
According to the Ambrosetti-Prodi theorem, the map $F(u)= - \Delta u - f(u)$ between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function $f$ is required. We show in two different ways that,…
We study a class of affine manifolds equipped with a flat affine connection $\nabla$ and a global Riemannian metric $g$ that is diagonal in local affine coordinates. These structures are closely related to \emph{Hessian manifolds}, where…
A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not…
A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($\Sigma^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of…
Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local…
The gauge invariant elastic metric on the shape space of surfaces involves the mean curvature and the normal deformation, i.e. the sum and the difference of the principal curvatures $\kappa_1,\kappa_2$. The proposed gauge invariant elastic…
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_p M) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we…
Irreducible representations are the building blocks of general, semisimple Galois representations \rho, and cuspidal representations are the building blocks of automorphic forms \pi of the general linear group. It is expected that when an…