arXiv research

Characterizes geodesics on spheres with Morse index bounds and inequalities.

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

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

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.

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

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.

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

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…

Paper calculates Morse index of Y-singular minimal surfaces.

Study on sphere immersions and their stability indices.

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

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.

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…

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.

Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.

Paper improves Morse index bound for hypersurfaces.

The paper proves index theorems for graph-based optimal control problems.

Proves a theorem for mechanical systems with reflections.

In this paper, we prove a Morse index theorem for the index form of even order linear Hamiltonian systems on the closed interval with reasonable self-adjoint boundary conditions. The highest order term is assumed to be nondegenerate.

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface. As a corollary, we find that for all immersed Willmore spheres $\vecΦ:S^2\rightarrow \math…

New Morse theory techniques glue nontransverse flowlines.

Global Morse index theorem applied to Jacobi fields on CMC surfaces.

We show that the rotationally symmetric free boundary minimal catenoid in the unit ball in $\Bbb{R}^3$ has Morse index equal to $4$.

We show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index.

The paper proves continuity of Morse index for Ricci shrinkers.

Study finds bound on energy of minimal spheres on complex manifolds.

We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.

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…

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-…

We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be ar…

We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing Riemannian metrics on S^4 that admit embedded minimal hyperspheres of uniformly…

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

Study bounds CMC surface index in 3-manifolds using energy.

The paper proves stability of critical points for conformally invariant Lagrangians.

Study Morse complexity of manifolds and homology classes, proving bounds and implications.

New proof for discrete Morse theory using combinatorial construction.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.