arXiv research

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

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.

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

Proves a theorem for mechanical systems with reflections.

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

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

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

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.

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

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…

Abstract Morse index theorem applied to various optimization problems.

We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in…

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…

Paper proves minimal hypersurface index for specific cases.

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints,…

We call a Morse function $f$ on a closed manifold $k$-constrained if neither $f$ nor $-f$ has critical points of indefinite Morse index $< k$. In this paper we study bordism groups of $k$-constrained Morse functions, and thus interpolate between the case $k = 1$ of bordism groups of Morse functions (computed by Ikegami…

This is essentially a note on Section 7 of Perelman's first paper on Ricci flow. We list some basic properties of the index form for Perelman's $\mathcal{L}$-length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse's index theorem for Perelman's $\mathcal{L}$-length holds…

We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of tho…

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

Given a Lorentzian manifold $(M,g)$, a geodesic $γ$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $γ$, we introduce a special class of instants along $γ$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate insta…

Rigidity theorem for critical points of Allen-Cahn equation on S³.

Main theorem of this paper states that Floer cohomology groups in a Hilbert space are isomorphic to the cohomological Conley Index. It is also shown that calculating cohomological Conley Index does not require finite dimensional approximations of the vector field. Further directions are discussed.

The paper calculates Morse indices and nullities for embedded networks on spheres.

Develops virtual Morse-Bott indices for four-manifolds, proving inequalities.

We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three…

Characterizes geodesics on spheres with Morse index bounds and inequalities.

The paper proves rigidity for mapping class group actions on metrics of positive scalar curvature.

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 …

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

We introduce Morse-type inequalities for a holomorphic circle action on a holomorphic vector bundle over a compact Kaehler manifold. Our inequalities produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomology in terms of the data of the fixed points and of the symplectic reduction. Th…

Paper proves finite Morse index for certain self-shrinkers.

Upper bound for Morse index of min-max varifolds.

Paper proves rigidity and index of Y-cones in unit ball.

A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex wi…

The paper proves the existence and properties of geodesics on convex surfaces.

We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homolog…

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

New constructions show stable geodesics and figure-eights in convex hypersurfaces.

Let $M$ be a compact surface and $P$ be a one dimensional manifold without boundary, that is the line $\mathbb{R}^1$ or a circle $S^1$. The classification of path-components of the space of Morse maps from $M$ into $P$ was recently obtained by S. V. Matveev and V. V. Sharko for the case $P=\mathbb{R}$. For $P=S^1$ the …

The paper proves the existence of certain minimal surfaces in specific manifolds.

We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on $\ls$. As an application we prove a theorem which can be interpreted as stating that this functional behaves "virtually" as a perfect Morse-Bott functional with a flow. This can be applied to study t…

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

In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We also analyze some natural Morse decompositions of the global attractor of the syst…

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

We prove a generalized version of the Morse index theorem for geodesics endowed with a non positive definite metric tensor (semi-Riemannian manifolds). We apply the result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of manifolds. More specifically, we c…

Study rational homology of moduli space via Morse functions, proving stability phenomena.

Paper calculates Morse index of Y-singular minimal surfaces.