arXiv research

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

We construct and analyze minimal disc stackings with bounds on their Morse index.

New spectral estimates for minimal surfaces with boundary conditions.

The paper bounds the Morse index and nullity of bipolar surfaces related to Otsuki tori.

Study of free boundary minimal surfaces with new existence and degeneration results.

Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula Ind^G(v) = \sum_{k = 0}^{n} (-1)^k χ^G(\d_k^+X) which takes its values in A(G). Here Ind^G(v) denotes the equiv…

The paper develops methods for calculating equivariant homology from Morse functions.

Study stabilizes Morse-Bott cohomology for equivariant manifolds.

In this paper, we prove equivariant Morse inequalities via Bismut-Lebeau's analytic localization techniques. As an application, we obtain Morse inequalities on compact manifold with nonempty boundary by applying equivariant Morse inequalities to the doubling manifold.

Generalizes Floer homotopy via Morse-Bott theory.

The paper studies Morse theory for Lie algebra actions on Riemannian foliations.

Characterizes geodesics on spheres with Morse index bounds and inequalities.

Study on minimal surfaces with Y-singularities, proving rigidity for Morse index one.

We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are allowed to "jump" along orbits of the S^1-action for finite time intervalls.

Study on Morse homology for reflection actions on manifolds.

Stability of Yang-Mills connections' Morse indices and nullity in 4D.

We investigate an equivariant generalization of Morse theory for a general class of integrable models. In particular, we derive equivariant versions of the classical Poincaré-Hopf and Gauss-Bonnet-Chern theorems and present the corresponding path integral generalizations. Our approach is based on equivariant cohomology…

Paper proves finite Morse index for certain self-shrinkers.

Upper bound for Morse index of min-max varifolds.

Explain Arnold's proof of the Morse index theorem using Maslov index.

We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivari…

Study bounds the Morse index of a special torus to 1.

Study of equivariant movie moves for involutive links.

We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.

Proves Morse index theorem for geodesics in conic Finsler manifolds.

Study on singularity behavior of mean curvature flow with bounded curvature and index.

Let $G$ be a torus and $M$ a compact Hamiltonian $G$-manifold with finite fixed point set $M^G$. If $T$ is a circle subgroup of $G$ with $M^G=M^T$, the $T$-moment map is a Morse function. We will show that the associated Morse stratification of $M$ by unstable manifolds gives one a canonical basis of $K_G(M)$. A key in…

Stability of Morse index for Yang-Mills connections in 4D.

Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline M$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal on $X$. We show that the $\overline\partial$-Neumann Laplacian on $M$ is transversally elliptic and as a conseque…

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes inf…

Study of harmonic oscillators on singular geometries using supersymmetry.

Paper calculates Morse index of Y-singular minimal surfaces.

Stability of Morse index for harmonic maps on degenerating surfaces analyzed.

Study on sphere immersions and their stability indices.

Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.

In this paper, we prove a Morse index theorem for the index form of regular Lagrangian system with selfadjoint boundary condition.

Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and …

The systole function has a universal index gap on moduli spaces.

The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.

Estimates the growth of Morse index for free boundary minimal hypersurfaces.

Abstract Morse index theorem applied to various optimization problems.

We give a new analytical proof of the Morse index theorem for geodesics in Riemannian manifolds.

We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse inde…

In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…

Low-index harmonic maps from $S^3$ to $S^2$ are simple.

We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs $(A,Φ)$, where $A$ is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and $Φ$ is a holomorphic section of $(E, d_A")$. We prove that a certain explicitly defined substratification of the Morse str…

The $2^{nd}$ 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 $\triangle_{A} +\frac{k_{g}}{4}$, hence it is finite.