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.
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…
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.
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.
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.
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.
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of…
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.
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.
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.
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.
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 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.
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.
For an immersed minimal surface in R3, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resol…
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.
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.
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.
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. …
Given a virtual link diagram D, we define its unknotting index U(D) to be minimum among (m,n) tuples, where m stands for the number of crossings virtualized and n stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram …
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.
Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover o…
Study on scalar curvature bounds and manifold topological complexity.
problem Understanding the topological complexity of manifolds with scalar curvature constraints.
method Introduced a small scale index theorem to establish bounds for Gromov's simplicial norm.
result Upper bound for Gromov's simplicial norm established in terms of scalar curvature, volume, and injectivity radius.
We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi-Schoen (Invent. Math. 1985) to higher dimensions.
Study on Hitchin index for cohomogeneity one nearly Kähler structures.
problem Characterizing Hitchin index for cohomogeneity one nearly Kähler structures.
method Variational characterization, Morse-like index, cohomogeneity one symmetry, ODE eigenvalue problem.
result Obtained non-trivial lower bounds on Hitchin index for specific structure.
Bounds on long-term returns of leveraged ETFs are given.
problem Uncertainty in long-term returns of leveraged ETFs.
method Quadratic bounds on log-returns based on daily log-returns of the underlying index.
result Sufficient conditions for outperformance and underperformance of leveraged ETFs.
We show that for an immersed two-sided minimal surface in R3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R3 of index 2, as conjectured by Choe. Moreover, we show that the index of a immersed two-sided mini…
For any closed Riemannian three-manifold, we prove that for any sequence of closed embedded minimal surfaces with uniformly bounded index, the genus can only grow at most linearly with respect to the area.
On any surface we give an example of a metric that contains simple closed geodesics with arbitrary high Morse index. Similarly, on any 3-manifold we give an example of a metric that contains embedded minimal tori with arbitrary high Morse index. Previously no such examples were known. We also discuss whether or not suc…
The paper bounds the energy index of harmonic Gauss maps on surfaces.
problem Bounding the energy index of harmonic Gauss maps on surfaces.
method Analyzes closed and complete non-compact surfaces in Lie groups with bi-invariant metrics.
result The energy index of the Gauss map is bounded by the topological genus.
The study examines the index of MOTS in Kerr-Newman-de Sitter spacetime and its relation to mass and charge.
problem Investigating the index of MOTS in Kerr-Newman-de Sitter spacetime.
method Analyzing the spatial cross section of the cosmological horizon in the Kerr-Newman-de Sitter spacetime, proving index bounds and establishing area-charge estimates.
result Established bounds on the index of MOTS and a connection between MOTS with index one and General Relativity.
The paper bounds the expectation of empirical processes indexed by Hölder classes.
problem Estimating the expectation of the supremum of empirical processes for distributions on bounded sets.
method Providing upper bounds on the expectation of the supremum of empirical processes indexed by Hölder classes.
result Deriving non-asymptotic risk bounds for estimating distributions using empirical processes and IPM.