New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
The study sets constraints on 4-manifold forms linked to specific invariants.
problem Understanding the intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres.
method Analyzes constraints using the μ-bar and κ invariants.
result The difference between κ and -μ-bar for a Seifert rational homology 3-sphere is at most 2, and under certain conditions, it is 0.
New proof of Sobolev inequality with constraints on sphere.
problem Improving Sobolev inequality on sphere with constraints.
method Careful study of extremal problem on sphere.
result Explicit determination of constant in second order moments case.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
Sphere theorems for specific manifolds with curvature constraints.
problem Sphere theorems for Riemannian manifolds with scalar curvature bounds and non-collapsed RCD(n−1,n) spaces. method Analysis of scalar curvature and mean distance constraints.
result Established sphere theorems for the specified manifolds.
The paper examines rigidity of special submanifolds in spheres with curvature constraints.
problem Rigidity of k-extremal submanifolds in a sphere under curvature conditions. method Proves pinching theorems for submanifolds with various curvature conditions.
result Various curvature conditions lead to rigidity of k-extremal submanifolds. In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The linearisation of the resulting fully nonlinear PDE is used to prove a short time…
New inequality criterion for a mean field equation on spheres.
problem Finding uniqueness in a mean field equation on spheres.
method Established a new Moser-Trudinger-Onofri inequality with a constraint on moments deviation.
result A threshold for deviation is a uniqueness criterion for the mean field equation.
Improved Moser-Trudinger-Onofri inequality with constraints on sphere.
problem Improving constant in Moser-Trudinger-Onofri inequality with constraints.
method Generalization to higher order moments and perturbation method.
result New inequalities with similarity to Lebedev-Milin type inequalities.
Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.
problem Cosmetic surgeries on knots in homology spheres and their constraints.
method Rational surgery formula of the Casson-Walker invariant for 2-component links.
result Constraints on knots and surgery slopes for cosmetic surgeries.
Two spheres found with specific curvature constraints.
problem Existence of spheres with prescribed mean curvature.
method Proved existence of at least two embedded spheres with curvature h satisfying pinching condition. result Existence of at least two embedded spheres with prescribed mean curvature h. In this paper, we show that, under arbitrary bounded Willmore energy assumption, embedded Willmore spheres (or more generally, embedded Willmore spheres under area constraint) with small diameter in a given 3-dimensional Riemannian manifold (M,h) necessarily concentrate at a critical point of the scalar curvature …
Manolescu correction terms are numerical invariants of homology three-spheres arising from Pin(2)-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere…
In relation to the 4-dimensional smooth Poincaré conjecture we construct a tentative invariant of homotopy 4-spheres using embedded contact homology (ECH) and Seiberg-Witten theory (SWF). But for good reason it is a constant value independent of the sphere, so this null-result demonstrates that one should not try to us…
Proves constraints on groups extending Möbius transformations on spheres.
problem Constraints on groups extending Möbius transformations on spheres.
method Proved constraints through group transitivity and topological entropy analysis.
result Groups must be 4-transitive or arc 4-transitive, and contain elements of positive topological entropy.
The paper finds geodesics on specific Finsler spheres with unique properties.
problem Identifying geodesics on Finsler spheres with given curvature constraints.
method Analyzes Finsler 4-spheres with specific curvature conditions to determine geodesic properties. result Proves existence of at least four prime closed geodesics under certain conditions.
Study symplectic cobordisms to empty sets, called hats, in 3-sphere and blow-ups.
problem Constraints on fillings of 3-sphere double covers.
method Relative symplectic cobordisms, focusing on 3-sphere and blow-ups of CP^2.
result Constraints on algebraic topology of fillings.
In this article we give a criterion for the existence of a metric of curvature 1 on a 2-sphere with n conical singularities of prescribed angles 2πϑ1,…,2πϑn and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dot…
A novel method relaxes binary constraints to non-negative spheres for multi-matching and clustering.
problem Optimization problems over binary matrices with injectivity constraints.
method Non-negative spherical relaxation followed by conditional power iteration.
result Automatic adjustment of the continuous parameter related to universe size.
Study minimizes Willmore energy with constraints on surface properties.
problem Minimizing Willmore energy under specific surface properties.
method Adapting Keller-Mondino-Rivière, Bauer-Kuwert, and Ndiaye-Schätzle methods.
result Existence of smooth minimizers for a broad range of constraints.
Statistical models with constrained probability distributions are abundant in machine learning. Some examples include regression models with norm constraints (e.g., Lasso), probit, many copula models, and latent Dirichlet allocation (LDA). Bayesian inference involving probability distributions confined to constrained d…
Method optimizes knotting pathways in constrained polymers.
problem Understanding how geometric constraints affect knot formation in polymers.
method Topological steering using knotoid spectrum and mean unravelling number.
result Geometric constraints increase the frequency of twist knots in polymers.
Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.
problem Contact cosmetic surgeries for Legendrian knots in L-spaces.
method Adapting techniques from S3 to L-spaces, incorporating Heegaard Floer theory constraints.
result Contact cosmetic surgery conjecture holds for non-trivial Legendrian knots, except for Lagrangian slice knots.
Equations link metrics with tensors, revealing curvature constraints.
problem Understanding curvature properties of geometric structures.
method Formal analogies to Einstein-Maxwell equations, studying Codazzi and conformal Killing equations.
result Constraints on scalar curvature of metrics in solutions.
The paper studies hyperbolic three-manifolds and their geometric constraints.
problem Understanding the interaction between hyperbolic geometry and homology cobordism.
method Derived explicit bounds on relative grading and invariants in monopole Floer homology.
result Explicit bounds on numerical invariants and subgroup structure of homology cobordism.
Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
We prove in this article that the local image of each conformal Q-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.
Delaunay tori minimize Willmore energy under isoperimetric constraints.
problem Finding minimizers of the Willmore energy under isoperimetric constraints.
method Constructing Delaunay tori using complete elliptic integrals and analyzing their Willmore energy.
result Existence of smoothly embedded tori minimizing the Willmore functional under isoperimetric constraints.
Study geometric properties and topology of curves on a sphere with curvature constraints.
problem Characterize curves on a sphere with specified curvature bounds.
method Introduced concepts of lower and upper curvatures, proved topological equivalence, and provided a differential equation solution.
result Proved curves in Pκ1κ2(P,Q) are uniquely determined by solutions to a specific differential equation. The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs…
The paper proves a theorem linking ball volume and Uryson width, with applications to 3-sphere metrics.
problem Relating volume of balls to Uryson width and Hausdorff content.
method Short proof and generalization of Guth's theorem, using Riemannian metrics and map diameter constraints.
result For any C>0, there exists a 3-sphere metric with volume 1 and a map violating diameter constraints. New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.
problem Optimizing mean curvature in Heisenberg group sub-Riemannian setting.
method Developed variational theory, established first and second variation formulas, introduced new critical surfaces.
result Identified and characterized a new family of rotationally invariant critical surfaces, the Pansu-Minkowski spheres.
Spheres minimize weighted curvature on spheres.
problem Minimizing weighted mean curvature on spheres.
method Solving a limit of Euler-Lagrange equations for Lp problems. result Solutions have at most three mean curvature values.
Study on contact type hypersurfaces in 4-space, proving no Brieskorn spheres can embed.
problem Constraints on the topology of 3-manifolds as contact type hypersurfaces in R4. method Obstruction derived from Heegaard Floer homology.
result No Brieskorn homology sphere can be embedded in R4. We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…
We show that, if a rational homology 3-sphere Y bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by Y. To this end, we make use of constraints on definite…
Second paper applies Morse index to constrained optimization problems.
problem Optimization problems with constraints on capillary surfaces.
method Abstract Morse index formulation applied to capillary surfaces.
result Precise determination of indices with constraints for various examples.
Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
We investigate constraints on embeddings of a non-orientable surface in a 4-manifold with the homology of M×I, where M is a rational homology 3-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó d-invariants or …
Study Lefschetz fibrations with 4D fibers using Seiberg-Witten theory.
problem Constraints on the topology of Lefschetz fibrations with 4D fibers.
method Using the family Bauer-Furuta invariant and framed bordism class of 1D Seiberg-Witten moduli spaces.
result New obstructions to the smooth isotopy of compositions of Dehn twists.
We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…
Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in S3 is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-po…
The study classifies certain types of incomplete surfaces with low curvature.
problem Classifying incomplete affine spheres with specific curvature constraints.
method Analyzing total curvature and asymptotic behavior of surfaces.
result New examples of incomplete affine spheres with positive genus found.
We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a ge…
New formalism solves kinematical constraints in curved backgrounds and non-trivial states.
problem Solving kinematical constraints due to Weyl invariance in curved backgrounds and non-trivial states.
method Constructing Weyl covariant geometric objects and identifying them as building blocks of correlation functions.
result Exact agreement with thermal OPEs and holographic computations for thermal 2-point functions.
New method solves sparse deconvolution problems with theoretical guarantees and practical applications.
problem Extracting localized, recurring motifs in signals with spatial or temporal structure.
method Geometric approach using sphere constraints and data-driven initialization to derive a provable algorithm.
result Practical algorithm solves real-world deconvolution problems with good performance and generalizability.
If a knot K bounds a genus one Seifert surface F in the 3-sphere and F contains an essential simple closed curve alpha that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice, then there are strong constraints on the algebraic concord…