Separation by a codimension-1 map with a normal crossing point

Letf:M ^n–1N ⁿ be an immersion with normal crossings of a closed orientable (n–1)-manifold into an orientablen-manifold. We show, under a certain homological condition, that iff has a multiple point of multiplicitym, then the number of connected components ofN–f(M) is greater than or equal tom+1, generalizing results of Biasi and Romero Fuster (Illinois J. Math. 36 (1992), 500–504) and Biasi, Motta and Saeki (Topology Appl. 52 (1993), 81–87). In fact, this result holds more generally for every codimension-1 continuous map with a normal crossing point of multiplicitym. We also give various geometrical applications of this theorem, among which is an application to the topology of generic space curves.     

Minimal sphere bundles in Euclidean spheres

The purpose of this paper is to pursue to work initiated by Hsiang-Lawson and study cohomogeneity 1 minimal hypersurfaces in Euclidean spheres which are equivariant under the linear isotropy representation of a rank 3 compact symmetric space.Supported by the grant NSF DMS 90-01089 and by CNPq (Brazil)     

Great circular surfaces in the three-sphere

In this paper, we consider a special class of the surfaces in the 3-sphere defined by one-parameter families of great circles. We give a generic classification of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry.     

Mirror duality in a Joyce manifold

Previously the two of the authors defined a notion of dual Calabi-Yau manifolds in a G₂ manifold, and described a process to obtain them. Here we apply this process to a compact G₂ manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror duals of each other.     

On caustics of submanifolds and canal hypersurfaces in Euclidean space

We investigate a relationship between the caustics of a submanifold of general dimension and of a canal hypersurface of the submanifold in Euclidean space. As a consequence, these caustics are the same. Moreover, induced Lagrangian immersion germs are Lagrangian equivalent under a suitable condition. In order to show the results, we use the theory of Lagrangian singularity and of Legendrian singularity.     

Extensions of the mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space

In this paper we define one-parameter families of Legendrian double fibrations in the products of pseudo-spheres in Lorentz-Minkowski space which are the extensions of four Legendrian double fibrations in the previous research (Izumiya, 2009 [9]). We show that these are contact diffeomorphic to each other. Moreover, we construct one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in these pseudo-spheres as applications of such extensions of the Legendrian double fibrations.     

Classification of surfaces in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map

In this article, we study submanifolds in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map. We give a characterization theorem for Lorentzian surfaces in the pseudo‐sphere with zero mean curvature vector in and 2‐type pseudo‐spherical Gauss map. We also prove that non‐totally umbilical proper pseudo‐Riemannian hypersurfaces in a pseudo‐sphere with non‐zero constant mean curvature has 2‐type pseudo‐spherical Gauss map if and only if it has constant scalar curvature. Then, for we obtain the classification of surfaces in with 2‐type pseudo‐spherical Gauss map. Finally, we give an example of surface with null 2‐type pseudo‐spherical Gauss map which does not appear in Riemannian case, and we give a characterization theorem for Lorentzian surfaces in with null 2‐type pseudo‐spherical Gauss map.     

How tightly can you fold a sphere?

Consider a compact, connected Lie group G acting isometrically on a sphere Sn of radius 1. The quotient of Sn by this group action, Sn/G, has a natural metric on it, and so we may ask what are its diameter and q-extents. These values have been computed for cohomogeneity one actions on spheres. In this paper, we compute the diameters, extents, and several q-extents of cohomogeneity two orbit spaces resulting from such actions, and we also obtain results about the q-extents of Euclidean disks. Additionally, via a simple geometric criterion, we can identify which of these actions give rise to a decomposition of the sphere as a union of disk bundles. In addition, as a service to the reader, we give a complete breakdown of all the isotropy subgroups resulting from cohomogeneity one and two actions.     

The Gauss map of submanifolds in the Heisenberg group

We obtain criteria for the harmonicity of the Gauss map of submanifolds in the Heisenberg group and consider the examples demonstrating the connection between the harmonicity of this map and the properties of the mean curvature field. Also, we introduce a natural class of cylindrical submanifolds and prove that a constant mean curvature hypersurface with harmonic Gauss map is cylindrical.     

Localization and duality in topology and modular representation theory

We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.Let G be a finite group and k be an algebraically closed field of characteristic p. If p is a homogeneous nonmaximal prime ideal in H^∗(G,k), then there is an idempotent module κ_p which picks out the layer of the stable module category corresponding to p, and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59-80] in their development of varieties for infinitely generated kG-modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of H^∗(G,k)/p as a graded H^∗(G,k)-module. Since κ_p can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_p is the p-completion of cohomology , and the statement that κ_p is a pure injective kG-module.In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357-402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to .The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^∗(BG;k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^∗(BG;k).In a companion paper [D.J. Benson, Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744-1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.     

On a new embedding theorem and the CLR-type inequality for Euclidean and hyperbolic spaces

The goal of this note is to provide a new embedding theorem and to derive from this embedding the CLR-type inequality for a potential belonging to a proper subspace of integrable functions.     

Contraction of convex hypersurfaces in Euclidean space

We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.This work was carried out while the author was supported by an Australian Postgraduate Research Award and an ANUTECH scholarship.     

The Hopf invariant of a Haefliger knot

We show that for any given differentiable embedding of the three-sphere in six-space there exists a Seifert surface (in six-space) with arbitrarily prescribed signature. This implies, according to our previous paper, that given such a (6,3)-knot endowed with normal one-field, we can construct a Seifert surface so that the outward normal field along its boundary coincides with the given normal one-field. This aspect enables us to understand the resemblance between Ekholm–Szűcs’ formula for the Smale invariant and a formula in our previous paper for differentiable (6,3)-knots. As a consequence, we show that an immersion of the three-sphere in five-space can be regularly homotoped to the projection of an embedding in six-space if and only if its Smale invariant is even. We also correct a sign error in our previous paper: “A geometric formula for Haefliger knots” [Topology 43: 1425–1447 2004].      

A minimizing deformation of Legendrian submanifolds in the standard sphere

In this note we study the moduli space of minimal Legendrian submanifolds in the standard sphere S²ⁿ⁻¹. We show that new examples of minimal Legendrian submanifolds can be constructed, if we can solve a certain equation for a function on a nearby glued Legendrian submanifold. As a step toward solving this equation, we prove short-time existence for a particular gradient flow on the space of immersed Legendrian submanifolds. A new necessary condition for a Lagrangian embedding into is given.     

Sommets et normales concourantes des courbes convexes de largeur constante et singularités des hérissons

LetC be a convex curve of constant width and of classC ₄ ⁺ . It is known thatC has at least 6 vertices and its interior contains either a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then: (i)C has exactly 6 vertices if, and only if, its evolute is the boundary of a topological disc through each interior point of which pass at least 6 normals; (ii) ifC has more than 6 vertices, then there exists an open set of points through each of which pass at least 10 normals. The proof: (i) expresses the number of normals passing through a point as a function of the index with respect to the evolute; (ii) relates this index to the number of singularities of the evolute (i.e. of vertices). Furthermore, we give formulas for counting singularities of generic hedgehogs in ℝ² and ℝ³.

     

Singular fibers and characteristic classes

Let be a proper generic map between smooth manifolds with dimN−dimM=−1. We explicitly calculate the cohomology class dual to the closure of the set of points in N over which lies a specific singular fiber in terms of characteristic classes of M and N.     

Hyperbolic geometry and local Dirichlet-Neumann map

We show in three dimensions that if we know the Dirichlet-to-Neumann (DN) map associated to the Schrödinger equation with a potential around the intersection of a sphere centered outside the convex hull of a bounded domain with smooth boundary, we can determine uniquely the corresponding spherical mean of the potential with an appropriate measure. This allows one to determine the potential near the boundary where the DN map is measured. Similar results are valid for electrical impedance tomography.     

Algebraic curves and the Gauss map of algebraic minimal surfaces

In this paper, we first establish a second main theorem for algebraic curves into the n-dimensional projective space. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in Rm with finite total curvature, as well as the uniqueness problem.