The paper proves strong holomorphic Morse inequalities on complex manifolds with optimal estimates.
problem Holomorphic Morse inequalities on non-compact complex manifolds with optimal fundamental estimates.
method Established strong holomorphic Morse inequalities under optimal fundamental estimates.
result Strong holomorphic Morse inequalities hold true on non-compact complex manifolds with optimal fundamental estimates.
The Morse-Smale complex of a function f decomposes the sample space into cells where f is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on es…
The paper proves Morse estimates for translated points on unit tangent bundles.
problem Estimating the minimal number of translated points in unit tangent bundles.
method Analyzing contactomorphisms of SM that lift diffeomorphisms of M homotopic to identity. result Proves the existence of sequences (pn,tn) with tno+∞ for a large class of manifolds. Estimates the growth of Morse index for free boundary minimal hypersurfaces.
problem Estimating the Morse index of free boundary minimal hypersurfaces.
method Adapting Song's method to the free boundary case.
result The Morse index grows linearly with the sum of Betti numbers.
Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper. Also we get an estimation on Arnold's question on a number of critical points of a Morse function with given boundary condition.
The ambient framed bordism class of the connecting manifold of two consecutive critical points of a Morse-Smale function is estimated by means of a certain Hopf invariant. Applications include new examples of non-smoothable Poincare duality spaces as well as an extension of the Morse complex.
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
The 2nd variation formula of the Seiberg-Witten functional is obtained in order to estimate the Morse index of redutible solutions (A,0). It is shown that their Morse index is given by the dimension of the largest negative eigenspace of the operator △A+4kg, hence it is finite.
The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.
problem Optimal estimates and inequalities for spectral functions on weakly 1-complete manifolds.
method Establishes optimal fundamental estimates and weak Morse inequalities for lower energy forms.
result Optimal fundamental estimates and weak Morse inequalities are proven for lower energy forms on weakly 1-complete manifolds.
The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. We advance the theory further and prove the first general Morse index bounds for minimal hypersurface…
We discuss the Morse estimates for the curvature of several metrics on Semple weighted projective bundle over a projective variety. Following Demailly works on holomorphic Morse inequalities we show an analogue of his results along the Green-Griffiths conjecture for invariant jets.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
New spectral estimates for minimal surfaces with boundary conditions.
problem Quantifying the Morse index of free boundary minimal surfaces.
method Adapted Montiel-Ros partitioning methods to compact manifolds with boundary, accounting for mixed and group actions.
result Explicit two-sided linear bounds on the Morse index for minimal surfaces.
Study on 4D Riemannian manifolds solves curvature problem.
problem Resonant prescribed T-curvature problem on compact manifolds.
method Variational theory, energy and gradient estimates, Morse lemma, Liouville technique.
result New existence results for critical points at infinity.
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.
We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage…
This paper improves a local-to-global principle for Morse quasigeodesics.
problem Quantify the size of local neighborhoods for global Morse behavior.
method Estimates in symmetric space to supplement Kapovich-Leeb-Porti's proof.
result Explicit criteria for local-to-global principle verified.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of G-invariant minimal hypersurfaces. method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
Let X be a hermitian manifold and let L^k be a high power of a hermitian line bundle over X. Local versions of Demailly's holomorphic Morse inequalities are presented - after integration they yield the usual inequalities. The local weak inequalities hold on any hermitian manifold X, regardless of compactness and comple…
The paper develops Morse homology for a class of elliptic partial differential equations.
problem Developing Morse homology for elliptic partial differential equations.
method Introducing a new notion of non-degeneracy and proving it generically satisfied for a class of functionals defined on Banach spaces.
result The paper enlarges the class of elliptic pde's for which non-degeneracy holds and Morse homology can be defined.
The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…
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.
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…
Study Witten deformation on noncompact manifolds with bounded geometry.
problem Cohomology of Witten deformation on noncompact manifolds.
method Witten deformation, Agmon estimate, Witten's instanton complex.
result Cohomology of Witten deformation is isomorphic to Thom-Smale and relative cohomology.
Paper compares higher torsions and removes fiberwise Morse function assumption.
problem Comparing higher torsions from analytic and topological perspectives.
method Introduced fiberwise generalized Morse functions (GMFs) and excised neighborhoods around birth-death points.
result Established a generalized version of the higher Cheeger-Müller/Bismut-Zhang theorem.
We prove that finite Morse index solutions to the Allen-Cahn equation in R2 have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularit…
For all n, we define the n-dimensional critical catenoid Mn to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in Rn+1. We show that the Morse index MI(n) of Mn satisfies the following asymptotic estimate as …
The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for C0-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we…
Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.
problem Analyzing the Morse index and eigenvalues of free-boundary CMC hypersurfaces in the upper hemisphere.
method Proved results using the norm squared of the second fundamental form and eigenvalue estimates.
result Proved bounds on Morse index and eigenvalues for free-boundary CMC hypersurfaces.
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 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.
New symmetric Willmore tori emerge from Clifford torus in Berger spheres.
problem Finding new symmetric Willmore surfaces from Clifford torus.
method Applying bifurcation theory to estimate Morse index of Willmore surfaces.
result New symmetric Willmore tori emerge from Clifford torus.
We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated clo…
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.
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.
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for i…
The paper introduces Morse theory for Lie groupoids and proves inequalities.
problem Defining Morse theory for Lie groupoids and studying their properties.
method Introducing Morse Lie groupoid morphisms and proving their Morita invariance.
result Established Morse theory for Lie groupoids and proved Morse inequalities.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
problem Mapping discrete Morse functions to shellings for topological analysis.
method Inducing Morse shellings on the second barycentric subdivision of a simplicial complex.
result Critical tiles of induced shellings correspond to critical faces of the discrete Morse function.
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
problem Well-definedness of Morse-Bott-Smale chain complex.
method Unified five degeneracy relations into a single condition.
result Quasi-isomorphic to Morse-Smale-Witten chain complex, alternative proof of Morse Homology Theorem.
Random walk constructs Morse functions on surfaces.
problem Creating Morse functions on surfaces.
method Random walk method to construct Morse functions.
result Small set of Morse functions approximates any other function.
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.
The paper develops methods for calculating equivariant homology from Morse functions.
problem Calculating equivariant homology from equivariant Morse functions.
method Alter equivariant Morse functions to stable ones, use generic equivariant metrics, and analyze the Morse spectral sequence.
result Equivariant Morse functions induce a filtration that computes equivariant homology.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. 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.
We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
Classifies Morse boundaries of 3-manifold groups.
problem Classifying Morse boundaries of 3-manifold groups.
method Classifies Morse boundaries into 9 types based on geometric decompositions.
result 9 different homeomorphism types of Morse boundaries.