arXiv research

Paper proves finite Morse index for certain self-shrinkers.

Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.

Minimal hypersurfaces with specific properties are shown to be catenoids.

Finite index solutions to Bernoulli problem are always axially symmetric.

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.

Abstract Morse index theorem applied to various optimization problems.

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

The paper studies minimal hypersurfaces in R^n with finite total curvature and proves index bounds.

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…

New spectral estimates for minimal surfaces with boundary conditions.

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

The paper proves continuity of Morse index for Ricci shrinkers.

Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.

Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.

Study estimates self-shrinker index with conical ends, proving index bound.

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…

Study classifies solutions to a triharmonic Lane-Emden equation.

Ancient mean curvature flows start from unstable minimal hypersurfaces.

New method extends discrete Morse theory to simplicial complexes.

We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $Δu + f(u) = 0 \hbox{in} \R^3$. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that …

Stable and Morse subgroups coincide in mapping class groups.

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.

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

Characterizes geodesics on spheres with Morse index bounds and inequalities.

New hyperbolic 4-manifolds found with special functions.

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

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

Study finds a catenoid's Morse index is 4.

Upper bound for Morse index of min-max varifolds.

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

Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result w…

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

Study of complex manifolds with boundary using equivariant index theorems and Morse inequalities.

Sharp Morse index bound for 2k-ended solutions in R^2.

Method calculates Morse index of branched Willmore spheres in 3-space.

Study on geodesics proving index and intersection bounds, with examples of multiplicity.

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

New Morse index bounds for min-max minimal hypersurfaces solved multiplicity problem.

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.

Highly curved spaces have surfaces with many bumps.

Paper proves Morse index of certain minimal hypersurfaces equals their homology class dimension.

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…

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…

Paper calculates Morse index of Y-singular minimal surfaces.

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