Related papers: Stability for holomorphic spheres and Morse theory

We study the orbit behavior of a four dimensional smooth symplectic diffeomorphism $f$ near a homoclinic orbit $\Gamma$ to an 1-elliptic fixed point under some natural genericity assumptions. 1-elliptic fixed point has two real eigenvalues…

Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear Hodge heat…

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined…

Consider the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\om)$ with the Hofer $L^{\infty}$-norm. A path in $\Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of…

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus…

This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Sere and to Coti-Zelati, Ekeland and Sere,…

We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2…

Let $f:T^2\to\mathbb{R}$ be a Morse function on a 2-torus, $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be its stabilizer and orbit with respect to the right action of the group $\mathcal{D}(T^2)$ of diffeomorphisms of $T^2$,…

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with Gromov-Witten/Floer…

We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with infinite dimensional symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily…

We prove a homological stability theorem for certain complements of symmetric spaces. This is a variant of a conjecture by Vakil and Matchett Wood for subspaces of $\mathrm{Sym}^n(X)$ where $X$ is an open manifold admitting a boundary. To…

Let $\pi : X = \mathbb{P}_C(E) \longrightarrow C$ be a ruled surface over an algebraically closed field $k$ of characteristic 0, with a fixed polarization $L$ on $X$. In this paper, we show that pullback of a (semi)stable Higgs bundle on…

Nonexistence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let $(N,h)$ be a complete noncompact Riemannian manifolds. Assume the universal covering of $(N,h)$ admits a nonnegative…

We use algebraic arc complexes to prove a homological stability result for symplectic groups with slope 2/3 for rings with finite unitary stable rank. Symplectic groups are here interpreted as the automorphism groups of formed spaces with…

Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kaehler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable…

This survey presents some recent results by the authors and Polterovich on the topological properties of ruled symplectic manifolds. The bundle M \to P \to B that is associated with a ruled manifold has the group of Hamiltonian…

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, \gamma). This is the space consisting of triples, (F_{g,n}, \phi, f), where F_{g,n} is a Riemann surface of genus g and…

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…

The goal of this paper is to study the subspace of stability condition $\Sigma_{\mathcal{E}}\subset \mathrm{Stab}(X)$ associated to an exceptional collection $\mathcal{E}$ on a projective variety $X$. Following Emanuele Macr\`{i}'s…

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty}$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of…