Related papers: Natural Frobenius Submanifolds

We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening…

We define the notion of mixed Frobenius structure which is a generalization of the structure of a Frobenius manifold. We construct a mixed Frobenius structure on the cohomology of weak Fano toric surfaces and that of the three dimensional…

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat…

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this…

We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds). It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds. J. Heber has showed that under certain simple…

A stratified pseudomanifold is normal if its links are connected. A normalization of a stratified pseudomanifold $X$ is a normal stratified pseudomanifold $Y$ together with a finite-to-one projection $n:Y\to X$ satisfying a local condition…

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to…

In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal…

We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for…

This is a brief survey of recent works by Neil Trudinger and myself on the Bernstein problem and Plateau problem for affine maximal hypersurfaces.

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda…

The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.

We introduce the notion of mixed trTLEP-structures and prove that a mixed trTLEP-structure with some conditions naturally induces a mixed Frobenius manifold. This is a generalization of the reconstruction theorem of Hertling and Manin. As a…

We investigate the integrability of Fr\'{e}chet tangent distributions on Fr\'{e}chet manifolds. We introduce the local well-posedness Condition W for split tangent subbundles, which reduces the local integrability problem to solving initial…

A real representation of a finite group naturally determines a polytope, generalizing the well-known Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general…

The existence of closed hypersurfaces of prescribed curvature in semi-riemannian manifolds is proved provided there are barriers.

Let $A/K$ be an absolutely simple abelian surface defined over a number field $K$. We give unconditional upper bounds for the number of prime ideals $\mathfrak{p}$ of $K$ with norm up to $x$ such that $A$ has supersingular reduction at…

Given a semisimple Frobenius manifold, we construct a class of integrable deformations of its hierarchy of topological type. We show that these integrable deformations have polynomial tau-structures, and conjecture that for the…

Mumford defined a natural isomorphism between the intermediate jacobian of a conic-bundle over $P^2$ and the Prym variety of a naturally defined \'etale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used…

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,...,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among…