Morse Index Bound for Minimal Two Spheres

Given a closed manifold of dimension at least three, with non-trivial homotopy group $$\pi _3(M)$$ π 3 ( M ) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width.

Blowing-up Solutions for 2nd-Order Critical Elliptic Equations: The Impact of the Scalar Curvature

Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet’s condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in \{4,5\}$, our proof requires to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$.

Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

On the topological complexity of manifolds with abelian fundamental group

We find conditions which ensure that the topological complexity of a closed manifold M with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik–Schnirelmann category of the cofibre of the diagonal map Δ : M → M × M {\Delta:M\to M\times M} for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds.

Polytope Novikov homology

Let M be a closed manifold and $${\mathcal {A}} \subseteq H^1_{\mathrm {dR}}(M)$$ A ⊆ H dR 1 ( M ) a polytope. For each $$a \in {\mathcal {A}}$$ a ∈ A , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $${\mathcal {A}}$$ A . The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.

A min-max characterization of Zoll Riemannian metrics

We characterise the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.

Short survey on the existence of slices for the space of Riemannian metrics

We review the well-known slice theorem of Ebin for the action of the diffeomorphism group on the space of Riemannian metrics of a closed manifold. We present advances in the study of the spaces of Riemannian metrics, and produce a more concise proof for the existence of slices.

On the Maximal Rate of Convergence Under the Ricci Flow

We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution that converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.