arXiv research

Study families of Morse functions for manifolds with boundary.

We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…

Locally, isoperimetric problems on Riemannian surfaces are sub-Riemannian problems in dimension 3. The particular case of Dido problems corresponds to a class of singular contact sub-Riemannian metrics : metrics which have the charateristic vertor field as symmetry. We give a classification of the generic conjugate loc…

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …

Constructs Lorentzian manifolds from Riemannian conformal structures.

In this paper the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of ce…

Existence of non-Einstein, non-shrinking Ricci solitons on quaternionic and octonionic spaces.

The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.

The classical Godbillon-Vey invariant is an odd degree cohomology class that is a cobordism invariant of a single foliation. Here we investigate cohomology classes of even degree that are cobordism invariants of (germs of) 1-parameter families of foliations.

Constructing solutions to the heterotic G$_2$ system on specific types of manifolds.

In this paper we show that every three-dimensional closed hyperbolic manifold admits no locally geometric $1$-parameter family of closed minimal surfaces.

For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.

We prove the existence of a new 2-parameter family o$Δ$ of embedded triply periodic minimal surfaces of genus 3. The new surfaces share many properties with classical orthorhombic deformations of Schwarz' D surface, but also exotic in many ways. In particular, they do not belong to Meeks' five-dimensional family. Never…

In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case…

Study $G_2$-instantons on $\mathbb{R}^4 imes S^3$ with invariant properties.

Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…

In this paper we provide a full classification of complete translating graphs in $\mathbf{R}^3$. We also construct two $(n-1)$-parameter families of new examples of translating graphs in $\mathbf{R}^{n+1}$.

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the …

We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of…

For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally …

Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on complex Lie groups. Both flat and non-flat cases are discussed in detail.

We show the existence of 1-parameter families of non-periodic, complete, embedded minimal surfaces in euclidean space with infinitely many parallel planar ends. In particular we are able to produce finite genus examples and quasi-periodic examples of infinite genus.

Smooth family of G2-instantons over a Kummer construction.

We characterize helix surfaces (constant angle surfaces) in the special linear group $\mathrm{SL}(2,\r)$. In particular, we give an explicit local description of these surfaces in terms of a suitable curve and a 1-parameter family of isometries of $\mathrm{SL}(2,\r)$.

We consider smooth 1-parameter families of plane curves tangent to a semicubic parabola, when the curvature radius of their curves at the tangency point vanishes at the cusp point. We find the $\A$-normal form of these families, their envelopes and local patterns near the cusp. We obtain a new codimension 2 singularity…

New homomorphism from Khovanov homology gives slice genus bounds.

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\R^2$. We study the singular…

The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relation between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the…

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.

We provide an existence proof for two 1-parameter families of embedded triply periodic minimal surfaces of genus three, namely the tG family with tetragonal symmetry that contains the gyroid, and the rGL family with rhombohedral symmetry that contains the gyroid and the Lidinoid, both discovered numerically in the 1990…

We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on…

Study shows orderability of certain Dehn surgeries on a specific knot family.

We present a 1-parameter family of finite action solutions to the $S0(2,1)$ Hitchin's equations and explore some of its basic properties. For a fixed value of the parameter, the solution is smooth. We conclude by showing a multi-particle generalization of our basic solutions.

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relations…

Constructs instantons on a specific G_2-manifold limit.

The classical H surfaces of H. A. Schwarz form a 1-parameter family of triply periodic minimal surfaces (TPMS) that are usually described as close relatives to his more famous P surface. However, a crucial distinction between these surfaces is that the P surface belongs to a 5-dimensional smooth family of embedded TPMS…

Study Floer theory of hyperbolic three-manifolds using Dirac spectral flow.

Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.

We characterize helix surfaces in the Berger sphere, that is surfaces which form a constant angle with the Hopf vector field. In particular, we show that, locally, a helix surface is determined by a suitable 1-parameter family of isometries of the Berger sphere and by a geodesic of a 2-torus in the 3-dimensional sphere…

The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in 1989. The purpose of this paper is to compare these two families. In particular …

We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying…

We show that a 1-parameter family Ricci flow ancient solutions arises from the natural collapsings of the twistor space of positive quaternion Kähler manifolds. We use these ancient solutions to show that a positive quaternion Kähler manifold is isometric to one of the Wolf spaces.

A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit met…

Study bifurcations and local rigidity on flag manifolds for Yamabe solutions.

New 1-parameter family of ovals identified in 4d Ricci flow classification.

The paper connects bundle curvature to random zero currents.

The paper classifies second-order superintegrable systems with torsion and semi-degeneracy.

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--…