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.
The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.
problem Proving inequalities for four-manifolds of Seiberg-Witten simple type.
method Uses virtual Morse-Bott indices and Hirzebruch-Riemann-Roch Theorem.
result Proves positivity of virtual Morse-Bott indices, leading to inequalities.
The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtra…
Proves functions can have two indices differing by two.
problem Understanding Morse index for smooth functions.
method Analyzes functions in unit sphere and ball.
result Exists functions with two indices differing by two.
We describe the configuration space S of polygons with prescribed edge slopes, and study the perimeter P as a Morse function on S. We characterize critical points of P (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a num…
The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.
problem Understanding the topology of 3-manifolds with certain Morse-Smale diffeomorphisms.
method Analyzing the structure of fixed points and separatrices of diffeomorphisms in 3-manifolds.
result All supporting manifolds of these diffeomorphisms are homeomorphic to lens spaces.
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.
problem Restricting Morse indices of saddles in gradient-like flows.
method Analyzing invariant manifolds and their intersections for gradient-like flows.
result Morse indices of saddles are either 1 or n-1, no other indices possible.
Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in R7. We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.
The paper calculates Morse indices and nullities for embedded networks on spheres.
problem Computing Morse indices and nullities for embedded networks on spheres.
method Using the Dirichlet-to-Neumann map and properties of eigenvalues and eigenfunctions.
result For all stationary triple junction networks in S2, there is only one eigenvalue -1. New minimal surfaces can have huge area and index.
problem Existence of minimal hypersurfaces with large area and index.
method Analyzing bumpy closed Riemannian manifolds.
result Sequence of minimal hypersurfaces with arbitrarily large area and index.
The paper proves index theorems for graph-based optimal control problems.
problem Optimal control problems on graphs with constraints.
method Proves Morse index theorems for a broad class of variational problems on graphs.
result Formulas compute the difference of Hessians related to different graphs or boundary conditions.
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while…
Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M having fixed number of critical points of each index, moreover at least χ(M)+1 critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a …
We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are co…
Study spider mechanism configuration spaces using squared distance function.
problem Understand configuration spaces of spider mechanisms.
method Use Morse theory of squared distance function from body to fixed point.
result List and describe critical manifolds of squared distance function as products of polygon spaces.
New spectral invariants distinguish Joyce orbifolds from other manifolds.
problem Distinguishing Joyce orbifolds from other G2-structures. method Introducing and computing two new spectral invariants for Joyce orbifolds.
result These invariants are more effective than existing invariants for distinguishing Joyce orbifolds.
We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups O(n). We describe the critical loci of the quadratic trace function Tr(AXBXT) and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of A and $B…
Study on phase transitions on surfaces using Allen-Cahn equation.
problem Existence of critical points with specific nodal sets on surfaces.
method Analysis of Allen-Cahn functional on compact surfaces, focusing on nodal sets and energy.
result Existence of countable families of critical points with nodal sets converging to geodesics.
Heat flow on lens spaces settles into Morse functions with four critical points.
problem Understanding the behavior of heat flow on lens spaces.
method Analyzing the asymptotic spectral expansion of the heat flow.
result Generic heat evolutions on lens spaces \(L(p,q)\) with \(p\geq2\) and \(1\leq q\leq p/2\) tend to settle into Morse functions with exactly four critical points.
Minimal submanifolds in spheres can be produced via Clifford type minimal products, and their Morse indices and nullities are calculated.
problem Understanding the properties of minimal submanifolds in spheres via Clifford products.
method Analyzing the first eigenfunctions and Morse indices of minimal products of minimal submanifolds.
result The Morse index and nullity of the minimal product are calculated and shown for specific cases.
In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an n-periodic minimal surface in Rn. In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number…
In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but…
The Nirenberg problem yields multiple conformal metrics for a given scalar curvature function.
problem Finding multiple conformal metrics with a given scalar curvature on spheres.
method Morse theoretical methods and counting index formulae, leveraging subcritical approximation and blowing-up solutions.
result Arbitrarily many metrics can be found that are conformally equivalent to the standard sphere and have the desired scalar curvature.
Given a closed manifold N and a self-indexing Morse function f: N --> R with up to four distinct Morse indices, we construct a symplectic Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle, f_C : D(T*N) --> C, when f is real analytic. By construction, pi: E --> C comes w…
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.
A celebrated result due to Poincaré affirms that a closed non-degenerate minimizing geodesic γ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented …
We present an algorithm to generate synthetic datasets of tunable difficulty on classification of Morse code symbols for supervised machine learning problems, in particular, neural networks. The datasets are spatially one-dimensional and have a small number of input features, leading to high density of input informatio…
This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ:R×X→X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X. Any compact tame set admits plenty tame flows. We prove …
Study of critical points in Ginzburg-Landau approximation with stability results.
problem Stability of critical points in Ginzburg-Landau approximation.
method Application of previous joint method with T. Rivière for upper semi-continuity of extended Morse index.
result Upper semi-continuity of extended Morse index for sequences of critical points.
We consider the configuration space of planar n-gons with fixed perimeter, which is diffeomorphic to the complex projective space CPn−2. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …
Study SO(3) vortices over orbifold Riemann surfaces and monopoles.
problem Characterize solutions of SO(3) monopole equations.
method Use moduli spaces of SO(3) vortices over orbifold Riemann surfaces.
result Compute Morse-Bott indices of a function on moduli space.
Calculates spectral flow bounds for reducible solutions to Vafa-Witten equations.
problem Bounding spectral flow between diverging reducible solutions.
method Localization and excision techniques to calculate spectral flow.
result Bounds on spectral flow are given for reducible solutions.
The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of t…
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.
New method extends discrete Morse theory to simplicial complexes.
problem Discrete Morse theory on simplicial complexes.
method Morse shellings and compatible discrete Morse functions.
result Triangulated surfaces and manifolds have Morse shellable triangulations.
Paper constructs continuous families of topological Morse functions.
problem Existence and deformability of topological Morse functions.
method Simple construction of continuous families of topological Morse functions.
result Gives a construction of continuous families of topological Morse functions.
New curvature K(x) measures manifold properties without integrals.
problem Understanding curvature on compact Riemannian manifolds.
method Developed index expectation curvature K(x) for 2D manifolds, constructed as a product of sectional index expectation curvatures.
result For small 2D manifolds with boundary, definite sign index expectation curvature K(x) exists and satisfies Gauss-Bonnet relation.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
Introduces optimization geometrodynamics for dynamic geometric optimization.
problem Gradient-based optimization methods struggle with changing geometric constraints.
method Optimization geometrodynamics separates invariant and improvable geometric mismatches.
result Dynamic geometric complexity measures the minimum geometric cost to reduce optimization difficulty.
Given a compact four-dimensional Riemannian manifold (M,g) with boundary, we study the problem of existence of Riemannian metrics on M conformal to g with prescribed Q-curvature in the interior M˚ of M, and zero T-curvature and mean curvature on the boundary ∂M of M. This geometric …