Study on CMC hypersurfaces with bounded index and area, proving multiplicity one convergence and bounds on genus.
problem Understanding CMC hypersurfaces with bounded index and area.
method Bubble-compactness theory for embedded CMC hypersurfaces in low dimensions.
result Minimal blow-ups are all catenoids, and bounds on genus provided.
Bounds on the index of free boundary minimal surfaces.
problem Understanding the index of free boundary minimal surfaces.
method Comparison of energy and area indices, combining with previous work.
result Area index is bounded by a linear function of genus and boundary components.
Minimal surfaces with bounded index have linearly growing genus.
problem Understanding the growth of genus in minimal surfaces with bounded index.
method Analyzing sequences of closed embedded minimal surfaces in three-manifolds.
result The genus of minimal surfaces can only grow linearly with area.
Improved upper bound on superbridge index from 5b-3 to 3b-1.
problem Improving bounds on superbridge index for knots.
method Improved upper bound formula for superbridge index.
result Upper bound on superbridge index reduced from 5b-3 to 3b-1.
We calculate the alternating number of torus knots with braid index 4 and less. For the lower bound, we use the upsilon-invariant recently introduced by Ozsváth, Stipsicz, and Szabó. For the upper bound, we use a known bound for braid index 3 and a new bound for braid index 4. Both bounds coincide, so that we obtai…
Upper bound for Morse index of min-max varifolds.
problem Bounding Morse index of varifolds.
method Proving upper bound for Morse index of min-max stationary integral varifolds.
result Upper bound for Morse index of min-max stationary integral varifolds.
Study bounds index of minimal hypersurfaces in curved spaces.
problem Bounding the index of minimal hypersurfaces.
method Proved linear index bound using first Betti number and curvature.
result Index is bounded below by a linear function of first Betti number.
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast…
New upper bound found for arc index of spatial graphs.
problem Finding an upper limit for the arc index of spatial graphs.
method Extended the definition of arc presentation to spatial graphs and derived a new upper bound.
result The upper bound on the arc index of any spatial graph is lowest possible.
Study bounds the Morse index of a special torus to 1.
problem Bounding the Morse index of a conformal harmonic torus.
method Min-max construction with harmonic replacement and conformal harmonic torus.
result The Morse index is bounded by one.
We bound the index of a subgroup in iterated Kodaira fibrations.
problem Bounding the index of a subgroup in iterated Kodaira fibrations.
method Passing to a finite index subgroup of π_1(X) to achieve the desired structure.
result We provide a bound on the index of such a group.
Unified proof of knot unknotting bounds using Ma-Qiu index.
problem Finding bounds on the number of moves to unknot knots.
method Using the Ma-Qiu index to bound presentation distances and Gordian distances.
result Unified proof of various unknotting number bounds.
The paper calculates the index and nullity of Fraser-Sargent surfaces and provides bounds.
problem Computing the index and nullity of Fraser-Sargent surfaces.
method Analytical and numerical methods to compute index and nullity; provides bounds.
result Lower and upper bounds on the index of Fraser-Sargent surfaces inside the ball.
Improved bounds on minimal surfaces' genus and ends, resolving conjectures.
problem Bounding the genus and ends of minimal surfaces based on their index.
method Analyzing immersed minimal surfaces in R3 to derive new lower bounds. result Resolved several conjectures about minimal surfaces' classification.
Study bounds singular set of minimal hypersurfaces with index control.
problem Estimating singular set size of minimal hypersurfaces.
method Finite index and null singular set conditions on integral varifolds.
result Local measure bounds on singular set and upper Minkowski content.
New bound shows ropelength is at least as long as absolute braid index.
problem Finding a lower bound for the ropelength of a link.
method Defining absolute braid index and showing a constant factor lower bound.
result Ropelength is bounded below by absolute braid index.
Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
Paper extends danceability concept to knots with braid index bound.
problem Danceability of knot diagrams.
method Developed a new knot invariant using braid index.
result Danceability is bounded above by the braid index.
Paper tackles continual learning with single-index models, proving regret bounds.
problem Continual learning with single-index models across multiple tasks.
method Proposes a randomized strategy to learn a common single-index and task-specific link functions.
result Proves regret bounds for the proposed strategy under various loss function assumptions.
New upper bounds on superbridge index for 49 knots, increasing known results to 49.
problem Determining superbridge index for knots not covered by stick number.
method New upper bounds not derived from stick number.
result Superbridge index of 4 for 49 knots, increasing known results to 49.
Proves energy quantization for surfaces with bounded index.
problem Energy quantization for Willmore surfaces with bounded index.
method Translated the question to the conformal Gauss map's perspective and showed convergence in specific regions.
result Conformal Gauss map converges to a light-like geodesic in De Sitter space in neck or collar regions.
Extends index theorem to uniformly elliptic operators on manifolds.
problem Generalizing index theorem to uniformly elliptic operators.
method Local index theorem on manifolds of bounded geometry.
result Validates multigraded elliptic uniform pseudodifferential operators.
Paper proves bounds on knot indices using bisected vertex leveling of plane graphs.
problem Upper bounds on knot indices using plane graph embeddings.
method Bisected vertex leveling of plane graphs to prove upper bounds on braid index and arc index.
result Quadratic upper bound on minimal crossing number of delta diagrams.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
Study bounds CMC surface index in 3-manifolds using energy.
problem Bounding the index of CMC surfaces in 3-manifolds.
method Energy comparison to prove linear upper bound.
result Linear upper bound on CMC surface index.
The study shows products of certain polyhedra have a bounded index property.
problem Understanding the bounded index property for products of polyhedra.
method Analyzing sufficient conditions for the bounded index property in products of polyhedra.
result Products of closed Riemannian manifolds with negative sectional curvature have the bounded index property.
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
Study bounds the index of CMC surfaces in 3-manifolds.
problem Bounding the index of CMC surfaces in 3-manifolds.
method Proved index estimates for CMC surfaces in certain 3-dimensional submanifolds.
result Index of CMC surfaces is bounded below by a linear function of its genus when mean curvature is large enough.
The study finds robust index bounds for minimal hypersurfaces in specific geometric spaces.
problem Finding lower bounds on the index of minimal hypersurfaces.
method Analyzing minimal isoparametric hypersurfaces, Lie groups, and quaternionic Grassmannians.
result Robust index bounds are established that are linear in the first Betti number.
Study on minimal surfaces with free boundary in a half-space, improving index estimates.
problem Non-existence of index two embedded minimal surfaces with free boundary in a half-space.
method Improved estimates of Neumann and Dirichlet indices, simplified proof of lower bounds.
result Answered Ambrozio et al.'s question and provided new lower bounds.
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
problem Stability of Yang-Mills connections' Morse indices and nullity in 4D under weak convergence.
method Proves stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence.
result Stability of the sum of Morse indices and nullity of a sequence of Yang-Mills connections.
Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.
problem Degeneration of minimal hypersurfaces under stability or bounded index constraints.
method Analysis of sequences of minimal hypersurfaces, parameterization with controlled maps, and topological finiteness results.
result Minimal hypersurfaces can degenerate to singular ones with controlled geometry, topology, and singular set.
A new index measures how many changes are needed to turn virtual links into simple ones.
problem Determining the minimum number of changes needed to simplify virtual link diagrams.
method Defined the unknotting index using virtual crossings and classical changes, provided lower and upper bounds.
result Found bounds for the unknotting index of virtual links, including those from pretzel links.
Establishes a lower bound for Kähler-Einstein distance on certain domains.
problem Finding a lower bound for Kähler-Einstein distance on specific types of domains.
method Proves an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.
result Establishes a lower bound for the Kähler-Einstein distance on pseudoconvex domains with positive hyperconvexity index.
Develops methods to calculate global index of real polynomials.
problem Calculating the global index of real polynomials.
method Two methods: via atypical fibres and Milnor arcs clusters.
result Derives upper bounds for the global index, refining Durfee's degree-based bound.
New bounds on genus and area for CMC surfaces in 3-manifolds.
problem Bounding genus and area of CMC surfaces in 3-manifolds.
method Local degeneration of minimal surfaces and index-area bounds.
result Genus and area of CMC surfaces are bounded by index and area.
The study proves all prime knots up to 10 crossings have superbridge index ≤ 5.
problem Determining the maximum superbridge index for prime knots up to 10 crossings.
method New upper bounds on stick numbers and equilateral stick numbers for specific knots, leading to conclusions about superbridge index.
result All prime knots through 10 crossings have a superbridge index ≤ 5.
Study index bounds for harmonic maps sequences with bubbles.
problem Upper and lower bounds of index and nullity for harmonic maps.
method Study limiting behavior of eigenfunctions of linearized operator; diagonalize index form with bilinear form varying with sequence.
result Obtain index bounds and show convergence of eigenfunctions on weak limit, bubbles, and neck regions.
Paper bounds index of free boundary surfaces.
problem Estimating the index of free boundary surfaces.
method Proved Morse index lower bound as linear function of genus and boundary components.
result Morse index lower bound by linear function of genus and boundary components.
Uniformly proves index invariance for signature operators on manifolds.
problem Proving index invariance for signature operators under uniform homotopy.
method Uniform homotopy invariance of Roe index for signature operators.
result Uniform homotopy invariance of Roe index for signature operators.
Study finds bound on energy of minimal spheres on complex manifolds.
problem Finding bounds on energy of minimal spheres on complex manifolds.
method Proving existence of harmonic spheres with Morse index bound one.
result Sum of energies of minimal spheres realizes a geometric invariant width.
In 1968, Simons introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S^{n+1}. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The earliest results regarding the index focused on the case of minimal hypersurfaces. …
New knot invariant bounds crossing number and homology.
problem Bounding knot invariants and homology groups.
method Region crossing changes, Goeritz matrices.
result Minimum homology generators < multi-region index.
Study energy functional's second variation on manifolds, proving Morse index bounds.
problem Understanding the stability and index of phase transition interfaces.
method Extending gradient theory to minimal hypersurfaces, proving upper semicontinuity of stability operator eigenvalues.
result Upper bounds for the Morse index of limit interfaces without multiplicity or orientability conditions.
We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show …
Paper improves Morse index bound for hypersurfaces.
problem Improving Morse index bound for hypersurfaces.
method Construction of hierarchical deformations and restrictive min-max theory.
result Generalizes a result by X. Zhou for 3≤n+1≤7. Upper bound on index of rotationally symmetric self-shrinking tori.
problem Stability of singularities in mean curvature flow.
method Entropy functional and eigenvalue analysis.
result Upper bound on the index of rotationally symmetric self-shrinking tori.