The article disproves a local systolic inequality and shows a lower bound on filling area.
problem Proving a local systolic inequality and a lower bound on filling area.
method Analyzing Gromov's filling area conjecture and showing a computational mistake.
result The local systolic inequality was disproved and a lower bound on filling area was shown.
Proves upper bound on systolic ratio for circle fillings.
problem Bounding systolic ratio for circle fillings.
method Proved upper bound on systolic ratio depending on genus.
result Filling Area Conjecture holds for large genus.
This paper tackles Gromov's filling area conjecture using discrete graph theory.
problem Finding the smallest surface area for isometrically filling a circle.
method Using graph-theoretic tools like Menger's theorem to derive bounds.
result Discrete bounds translate to a lower bound of 1.36π for the hemisphere's surface area.
We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.
We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingr…
Improved upper bound for discrete isometric filling of cycles.
problem Finding the minimum number of vertices in a discrete isometric filling of cycle graphs.
method Explicit construction of isometric fillings using concentric annular structures.
result Explicit construction of isometric fillings with \( |V(K_n)| \le \left(\frac{1}{6} + o(1)
ight)n^2 \), improving the upper bound to \( D^* \le \frac{1}{6} \).
The Riemannian hemisphere has a lower bound for its mass.
problem Estimating the mass of surfaces spanning a circle.
method Constructing a differential form with a stationary comass norm on the hemisphere.
result The mass of surfaces spanning a circle has a lower bound of 2π plus a second-order term. The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.
problem Approximating Riemannian metrics and proving geometric conjectures.
method Discretization of metrics using walls and triangulations.
result The discrete filling area conjecture is equivalent to Gromov's original conjecture.
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…
The volume conjecture is proven for twist knots after Dehn filling.
problem Proving the volume conjecture for twist knots after Dehn filling.
method Constructing a new ideal triangulation of the Whitehead link complement.
result Chen-Yang's volume conjecture holds for sufficiently large parameters.
Researchers prove an inequality linking spin manifold fill-ins to hyperspherical radius.
problem Proving an inequality relating the infimum of mean curvature to hyperspherical radius for spin manifolds.
method Combining Hijazi-Montiel-Roldán inequality and Bär's recent theorem; providing an alternative proof based on Bär-Ballmann work.
result Proved inequality linking spin manifold fill-ins to hyperspherical radius.
Proves Gromov's conjecture on total mean curvature using surgery and positive mass theorems.
problem Proving Gromov's conjecture on total mean curvature of fill-ins.
method Surgery to reduce to fill-ins of spheres, positive mass theorems, and quantitative surgery process.
result Proves Gromov's conjecture on total mean curvature in various cases.
Estimates the degree of trace fields of hyperbolic Dehn fillings.
problem Estimating the complexity of hyperbolic 3-manifolds.
method Using Lehmer's conjecture, bounds the degree of trace fields.
result Estimates the degree of trace fields of hyperbolic Dehn fillings.
A pair (α,β) of simple closed geodesics on a closed and oriented hyperbolic surface Mg of genus g is called a filling pair if the complementary components of α∪β in Mg are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In \cite{Aou}, Aougab-Huang con…
Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.
problem Improving bounds on curve filling areas in non-geodesic Banach spaces.
method Improved bounds on curve filling areas in Banach spaces.
result Rigidity of Pu's classical systolic inequality.
Proves left-orderability of certain Dehn fillings on 3-manifolds.
problem Left-orderability of Dehn fillings on 3-manifolds.
method Constructing arcs of representations and using holonomy extension techniques.
result Establishes intervals of orderable Dehn fillings for odd pretzel knots.
Study L2-Betti numbers of Dehn fillings for special groups.
problem Investigate L2-Betti numbers of Dehn fillings. method Prove L2-Betti numbers equality for virtually special groups. result Verify Singer Conjecture for certain Einstein manifolds.
We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…
New group theory insights on knot surgery results.
problem Understanding non-simply connected 3-manifolds from Dehn surgery.
method Group theoretic analysis of Property P conjecture variations.
result New group theoretic perspectives on Dehn filling.
MO-GP models fill gaps in biophysical data with across-domain info transfer.
problem Gap filling of biophysical parameters LAI and fAPAR over rice areas.
method Multi-output Gaussian Processes (MO-GP) based on Linear Model of Coregionalization (LMC).
result MO-GP models successfully predict biophysical variables even in high missing data regimes.
New satellite knots counter a conjecture about Lorenz knots.
problem A conjecture about satellite knots and Lorenz knots was disproven.
method Constructed infinitely many counterexamples of satellite knots that are not cables.
result The conjecture was amended and shown to hold for many Lorenz knots.
The study computes trace fields and minimal polynomials for specific knots and links.
problem Computing trace fields and minimal polynomials for specific knots and links.
method Using factorization theorems for sparse polynomials.
result Results depend on the degrees of the trace fields over Q being sufficiently large.
This paper describes the complete list of all 205,822 exceptional Dehn fillings on the 1-cusped hyperbolic 3-manifolds that have ideal triangulations with at most 9 ideal tetrahedra. The data is consistent with the standard conjectures about Dehn filling and suggests some new ones.
Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order elements have a certain structure of a free product. We then apply this result to show that groups hyperbolic relative to residually finite groups satisfying the Farr…
Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.
problem Understanding which Seifert fibered spaces can be boundaries of symplectic rational homology balls.
method Analyzes convex boundaries and Lagrangian disk fillings of Legendrian knots.
result Strong restrictions on which Seifert fibered spaces can bound symplectic rational homology balls.
The paper solves a problem related to scalar curvature and boundary metrics.
problem Proving the extensibility of boundary metrics to positive scalar curvature metrics.
method Introducing a fill-in invariant and proving relationships with positive mass theorems.
result The positive mass theorem for asymptotically hyperbolic manifolds implies the same for asymptotically flat manifolds.
We show that Dehn filling on the manifold v2503 results in a non-orderable space for all rational slopes in the interval (−∞,−1). This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.
The Kauffman bracket skein module K(M) of a 3-manifold M is the quotient of the Q(A)-vector space spanned by isotopy classes of links in M by the Kauffman relations. A conjecture of Witten states that if M is closed then K(M) is finite dimensional. We introduce a version of this conjecture for ma…
Upper bound found for minimal area in Einstein 4-manifolds.
problem Finding the smallest area of a 2D varifold in closed Einstein 4-manifolds.
method Using homological filling functions and quantitative Sobolev and ε-regularity constants for Einstein metrics.
result An upper bound A(M,g)≤FEin(v,D) for the area of 2D varifolds in Einstein 4-manifolds. We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…
Proves a conjecture about metrics and minimal area enclosures.
problem Proving a conjecture about metrics and minimal area enclosures.
method Using boundedness of harmonic function u, proving the conjecture for asymptotically flat 3-manifolds.
result Proves the bounded conformal conjecture under the assumption of boundedness of harmonic function u.
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments whic…
In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free.'' We explain the relevance of this th…
This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.
New geometric methods solve a conjecture for hyperbolic 3-manifolds.
problem Verifying the 1-loop conjecture for hyperbolic 3-manifolds.
method Constructing geometric ideal triangulations and solving gluing equations.
result Proves the 1-loop conjecture for a large class of hyperbolic 3-manifolds.
The study confirms two cases of the convex body isoperimetric conjecture in the plane.
problem The least perimeter to enclose a given area inside a unit disk is greater than inside any other convex set.
method Examined symmetric domains and perturbations of the unit disk.
result Two cases of the convex body isoperimetric conjecture are confirmed.
New method shows some 3D shapes can't be filled in certain ways.
problem Obstructing Liouville and weak fillability of contact structures.
method Introducing a new method to obstruct fillability.
result Various rational homology 3-spheres admit strongly fillable contact structures without Liouville fillings.
We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for…
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.
problem Left-orderability of fundamental groups in Dehn fillings of pseudo-Anosov mapping tori.
method Two approaches: one using R-covered foliations and the other using one-sided branching. result All such Dehn fillings have left-orderable fundamental groups.
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.
3-manifolds are CR uniformized on spheres, proving a conjecture.
problem Uniformizing 3-manifolds with cusps using CR methods.
method Spherical CR uniformization of complex hyperbolic triangle groups.
result Magic 3-manifolds and other cusped 3-manifolds are CR uniformizable.
Proves volume conjectures for figure-eight knot surgeries.
problem Volume conjectures for hyperbolic 3-manifolds.
method Ohtsuki's method applied to figure-eight knot surgeries.
result Proves Asymptotic Expansion and Volume Conjectures for figure-eight knot surgeries.
An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call "Floer simple," we construct an invariant which computes the interval of L-space filling slopes from the Turaev torsion and a given slope …
The paper extends techniques to create non-left-orderable manifolds from links with multiple boundary components.
problem Creating infinite families of non-left-orderable manifolds from Dehn fillings.
method Order-detection of slopes to generalize techniques for multiple boundary components.
result Produces an infinite family of non-left-orderable Dehn fillings for hyperbolic links, including the Whitehead link.
Proves planar Lipschitz critical points of area functional are smooth.
problem Lawson-Osserman conjecture about smoothness of critical points.
method Outer variations to prove smoothness of critical points.
result Proves conjecture for planar case.
New findings show that not all area-minimizing surfaces are calibrated, even on complex manifolds.
problem Understanding when area-minimizing surfaces cannot be calibrated.
method Analyzing homology classes and metrics on manifolds to determine if area-minimizers are calibrated.
result Calibrated area-minimizers are non-generic, challenging the common assumption that they are typical.