arXiv research

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …

We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not tot…

On a compact, oriented, Riemannian manifold, the Hodge decomposition theorem associates a smooth primitive to any exact smooth form omega. In this paper, we show that given a smooth family of exact smooth forms omega(t), the family of associated primitives is also a smooth family with respect to t.

We present a 2x2 Lax representation for discrete circular nets of constant negative Gauß curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the membe…

We use the DPW method to obtain the associate family of Delaunay surfaces and derive a formula for the neck size of the surface in terms of the entries of the holomorphic potential.

We shall give a definition of the curvature operator for a family of weighted Bergman spaces $\{\mathcal H_t\}$ associated to a smooth family of smoothly bounded strongly pseudoconvex domains $\{D_t\}$. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature fo…

Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We v…

A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in …

The notion of a gerbe with connection is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to `characteristic gerbes' associated to families of bundles and connections. In turn th…

Three decades ago Cornalba-Harris proved a fundamental positivity result for divisor classes associated to families of stable curves. In this paper we establish an analogous positivity result for divisor classes associated to families of stable differentials.

Hildebrand classified all semi-homogeneous cones in $\mathbb{R}^3$ and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hil…

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the near…

We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space $\mathbb{R}^n$ and generalize in a natural way the notion of associated family. We show that the existence of such deformations turns out to be very restrictive and…

Study slopes of direct images in complex manifolds, proving a Mehta-Ramanathan type theorem.

Correspondence found between exponential families and affine Grassmannians.

We show how to associate with each graph with a certain property (positivity) a family of simply connected solvable Lie groups endowed with left-invariant Riemannian metrics that are Ricci solitons (called solsolitons). We classify them up to isometry, obtaining families depending on many parameters of explicit example…

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and o…

In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…

The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.

This paper gives two methods for constructing associative 3-folds in R^7, based around the fundamental idea of evolution equations, and uses these methods to construct examples of these geometric objects. The paper is a generalisation of the work by Joyce in math.DG/0008021, math.DG/0008155, math.DG/0010036 and math.DG…

New method constructs multiple group racks, differing from known constructions.

A model of associative memory is studied, which stores and reliably retrieves many more patterns than the number of neurons in the network. We propose a simple duality between this dense associative memory and neural networks commonly used in deep learning. On the associative memory side of this duality, a family of mo…

Computes quandle associated groups using group homology.

New polynomials defined for quandle structures, enhancing graph invariants.

Study on new Monge-Ampère functionals and their variational problems.

New theorem connects minimal and maximal surfaces, affecting graphness.

We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We …

An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…

Discuss Alan Schoen's I-WP minimal surface with geometric realizations.

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

This study addresses transitions in conically singular associative submanifolds and their desingularizations.

This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories…

Study on convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces.

In this paper we provide examples of maps from almost complex domains into pseudo-Riemannian symmetric targets, which are pluriharmonic and not integrable, i.e. do not admit an associated family. More precisely, for one class of examples the source has a non-integrable complex structure, like for instance a nearly Kaeh…

We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expresse…

Study on Alexander polynomials in braids, linking number theory and topology.

Let $M$ be a hyperkaehler manifold, and $η$ a closed, positive (1,1)-form which is degenerate everywhere on $M$. We associate to $η$ a family of complex structures on $M$, called a degenerate twistor family, and parametrized by a complex line. When $η$ is a pullback of a Kaehler form under a Lagrangian fibration $L$, a…

Constant mean curvature (CMC) surfaces in space forms can be described by their associated $\mathbb C^*$-family of flat $SL(2,\mathbb C)$-connections $\nabla^λ$. In this paper we consider the asymptotic behavior (for $λ\to0$) of the gauge equivalence classes of $\nabla^λ$ for compact CMC surfaces of genus $g\geq2.$ We …

We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given f…

The paper studies obstructions to solutions of the Wess-Zumino-Witten equation and its generalizations.

We consider families of Dirac operators on the unit interval which depend on parameters via boundary conditions. We study the associated eta forms and Maslov cocyles. With this simple example we show how previous results of Lesch/Woiciechowski and the first author on the eta invariant of cylinders generalize to the fam…

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pullbacks of hypergeometric functions by rational Belyi functions with restricted …

We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show …

Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required propert…

CAT(0) study of curve complements and families.