arXiv research

Study on curve diffusion flows with scale-critical curvature term.

We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.

In this paper we study the (asymptotic and exponential) stability of the $m$-fold circle as a solution of the $p$-curve shortening flow ($p\geq 1$ an integer).

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in pairs…

Sharp convergence rate for curvature stability in planar free elastic flow.

Knots in circle bundles are uniquely identified by their complements.

Study shows how a curve shortens to a half-circle under specific flow.

Wave maps from circle to manifold controllable if homotopy classes match.

We study the linear stability of Einstein metrics of Riemannian submersion type. First, we derive a general instability condition for such Einstein metrics and provide some applications. Then we study instability arising from Riemannian product structures on the base. As an application, we estimate the coindex of the E…

Let $(M,ω)$ be a Kähler manifold and let $K$ be a compact group that acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on probability measures on $M$. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystabili…

Stable actions of hyperbolic groups on their boundaries.

Paper calculates Morse index of Y-singular minimal surfaces.

The paper examines the stability of Killing cylinders in hyperbolic space.

Let $F$ be a leafwise hyperbolic taut foliation of a closed 3-manifold $M$ and let $L$ be the leaf space of the pullback of $F$ to the universal cover of $M$. We show that if $F$ has branching, then the natural action of $π_1(M)$ on $L$ is faithful. We also show that if $F$ has a finite branch locus $B$ whose stabilize…

We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show th…

The stability and robustness of compact schemes for parabolic PDEs are analyzed.

Defines new stability conditions for Sasaki manifolds and extremal metrics.

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…

Improved stability analysis of neural network systems using Zames-Falb multipliers.

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentu…

The study examines deformations of functions on surfaces.

We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). …

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.

The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms of homotopy equivalences of a graph fixing a subgraph. This is needed for the sec…

Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M) \times Diff(P)$ on the space of smooth mappings $C^{\infty}(M,P)$. For $f\in C^{\infty}(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these ac…

The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.

We give a topological stability result for the action of the fundamental group of a compact manifold of negative curvature on its boundary at infinity: any nearby action of this group by homeomorphisms of the sphere is semi-conjugate to the standard boundary action. Using similar techniques we prove a global rigidity r…

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case o…

New method shows how to move one Heegaard surface positioning to another.

The seriousness of the current crisis urgently demands new economic thinking that breaks the austerity vs. deficit spending circle in economic policy. The core tenet of the paper is that the most important problems that natural and social science are facing today are inverse problems, and that a new approach that goes …

A positive path in the linear symplectic group $\Sp(2n)$ is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-indepe…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of stabilizers and orbits of f with respect to the right action of the diffeomorphisms…

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular …

New invariants prove existence of Kahler-Einstein metrics on big classes.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these…

Study generates infinite circle packings with a specific property.

The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.

Solves Apollonius' problem using oriented circles and inversive geometry.

The paper explores universal circles for Anosov foliations and their uniqueness.

Link projections with the same circle arrangement can be transformed by specific moves.

Let $M$ be a smooth connected compact surface and $P$ be either a real line or a circle. This paper proceeds the study of the stabilizers and orbits of smooth functions on $M$ with respect to the right action of the group of diffeomorphisms of $M$. A large class of smooth maps $f:M\to P$ with isolated singularities is …

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 …

Proves rigidity of circle packings in the plane, generalizing previous work.

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending …

Classifies surfaces with great and small circles through each point.

Study local diffeomorphisms of conformal circles in pseudo-Riemannian manifolds.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.