arXiv research

The paper bounds the min-max width of embedded circles on spheres and manifolds.

In this note we establish estimates for the harmonic map heat flow from $S^1$ into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.

Given a Riemannian metric on the 2-sphere, sweep the 2-sphere out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show the following useful property (see Th…

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout …

Continuous sweepouts cover manifolds with bounded curve lengths.

Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than L, we construct a second sweepout composed of curves of length less than L which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Fre…

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$. The sweepo…

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted th…

Constructs a unique surface in a ball with specific properties.

Study finds a minimal surface in a ball with specific properties.

Closed Riemannian 4 or 5-manifolds contain branched immersed closed minimal surfaces.

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

The paper develops a theory for free boundary minimal surfaces with genus at least one.

New minimal surfaces found in 3D sphere with low genus.

We show that a free period three action on the three-sphere is standard, i.e. the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a resul…

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the …

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Co…

A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-di…

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of t…

The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. We advance the theory further and prove the first general Morse index bounds for minimal hypersurface…

We prove that for generic metrics on a 3-sphere, the minimal surface obtained from the min-max procedure of Simon-Smith has index 1. We prove an analogous result for minimal surfaces arising from strongly irreducible Heegaard sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that a strongly irr…

We show that if two 3-manifolds with toroidal boundary are glued via a `sufficiently complicated' map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by gluing X and Y, two connected small manifolds with incompressible boundary, along a closed sur…

We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each…

Study min-max theory for hypersurfaces with boundary constraints.

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …

Defines half-volume spectrum for manifolds and proves Weyl law holds.

We construct and analyze minimal disc stackings with bounds on their Morse index.

Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be tur…

Study compares two knot pairings and their equivalence.

The paper studies Fox pairings of Poincaré duality groups using group cohomology.

Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.

New examples of Cappell-Shaneson knot pairs with same Alexander polynomial found.

The paper constructs minimal coherent filling pairs on surfaces.

In this paper, we introduce some notions on the pair consisting of a Chern connection and a Higgs field closely related to the first and second variation of Yang-Mills- Higgs functional, such as strong Yang-Mills-Higgs pair, degenerate Yang-Mills-Higgs pair, stable Yang-Mills-Higgs pair. We investigate some properties …

The study of $PD_3$-pairs extends results for aspherical 3-manifolds.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

John Conway created pairs of domains that sound the same for a special kind of music.

As a vast number of ingredients exist in the culinary world, there are countless food ingredient pairings, but only a small number of pairings have been adopted by chefs and studied by food researchers. In this work, we propose KitcheNette which is a model that predicts food ingredient pairing scores and recommends opt…

This paper studies properties of weak reducing pairs in critical Heegaard splittings.

The paper studies the moduli space of Higgs pairs and their geometric properties.

We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of …

Explicit matrix presentations of Blanchfield pairings and twisted pairings for torus knots.

Pairs Trading is carried out in the financial market to earn huge profits from known equilibrium relation between pairs of stock. In financial markets, seldom it is seen that stock pairs are correlated at particular lead or lag. This lead-lag relationship has been empirically studied in various financial markets. Earli…

Pairs trading strategy improved using Ornstein-Uhlenbeck process.

Weak dual pairs defined in Dirac-Jacobi geometry, proving equivalence and leaf correspondence theorems.

A novel graphical matching approach improves pairs trading by reducing portfolio variance and risk-adjusted returns.