A Morse estimate for translated points of contactomorphisms of spheres and projective spaces

A point q in a contact manifold (M, ξ) is called a translated point for a contactomorphism ${\phi}$ with respect to some fixed contact form if ${\phi(q)}$ and q belong to the same Reeb orbit and the contact form is preserved at q. In this article we discuss a version of the Arnold conjecture for translated points of contactomorphisms and, using generating functions techniques, we prove it in the case of spheres (under a genericity assumption) and projective spaces.     

Contact structures on M \times S^2

We show that if a manifold $M$ admits a contact structure, then so does $M \times S^2$ . Our proof relies on surgery theory, a theorem of Eliashberg on contact surgery and a theorem of Bourgeois showing that if $M$ admits a contact structure then so does $M \times T^2$ .     

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

A group $G$ is said to be periodic if for every $g\in G$ there exists a positive integer $n$ with $g^n=\mathrm{Id}$ . We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a probability measure $\mu $ is finite. Moreover if the group consists of homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T }^2$ . In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.     

K?hler–Einstein metrics on strictly pseudoconvex domains

Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain ${M\subset X}$ in a complex manifold carries a complete K?hler–Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on ${\partial M}$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over ${\partial M}$ . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a K?hler–Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of K?hler–Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c ₁?<?0 admits a complete K?hler–Einstein metric.     

Locally conformally Kähler manifolds admitting a holomorphic conformal flow

A manifold M is locally conformally Kähler (LCK) if it admits a Kähler covering ${\tilde{M}}$ with monodromy acting by holomorphic homotheties. Let M be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on ${\tilde{M}}$ . We show that M admits an automorphic potential, and the monodromy group of its conformal weight bundle is ${\mathbb{Z}}$ .     

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

Every bordered Riemann surface is a complete proper curve in a ball

We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of $\mathbb C ^2$ , and a complete proper holomorphic embedding into a ball of $\mathbb C ^3$ .     

On the intersection forms of spin four-manifolds with boundary

We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from \(\mathrm{Pin }(2)\) -equivariant Seiberg-Witten Floer K-theory. In the process we introduce the notion of a Floer \(K_G\) -split homology sphere; this concept may be useful in an approach to the 11/8 conjecture.     

Homomorphism Bounds for Oriented Planar Graphs of Given Minimum Girth

We find necessary conditions for a digraph H to admit a homomorphism from every oriented planar graph of girth at least n, and use these to prove the existence of a planar graph of girth 6 and oriented chromatic number at least 7. We identify a ${\overleftrightarrow{K_4}}$ -free digraph of order 7 which admits a homomorphism from every oriented planar graph (here ${\overleftrightarrow{K_n}}$ means a digraph with n vertices and arcs in both directions between every distinct pair), and a ${\overleftrightarrow{K_3}}$ -free digraph of order 4 which admits a homomorphism from every oriented planar graph of girth at least 5.     

Holomorphic submersions of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold \(X\) admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then \(X\) is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.     

A sufficient condition for zero entropy

We prove that a diffeomorphism \(f\) defined on a compact manifold has zero topological entropy if there are \(d\in {\mathbb {N}}\) and \(K>0\) such that \(\Vert Dg^{n_x}(x)\Vert \le Kn^d_x\) for every diffeomorphism \(g\) that is \(C^1\) close to \(f\) and every periodic point \(x\) of least period \(n_x\) of \(g\) .     

Converting homotopies to isotopies and dividing homotopies in half in an effective way

We prove two theorems about homotopies of curves on two-dimensional Riemannian manifolds. We show that, for any \({\epsilon > 0}\) , if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by \({L + \epsilon}\) . If the manifold is orientable, then for any \({\epsilon > 0}\) we show that, if we can contract a curve \({\gamma}\) traversed twice through curves of length bounded by L, then we can also contract \({\gamma}\) through curves bounded in length by \({L + \epsilon}\) . Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.     

On the geometry of locally conformally almost cosymplectic manifolds

We obtain the complete group of structure equations of a locally conformally almost cosymplectic structure (an lc $ ACy $ -structure in what follows) and compute the components of the Riemannian curvature tensor on the space of the associated G-structure. Normal lc $ ACy $ -structures are studied in more detail. In particular, we prove that the contact analogs of A. Gray’s second and third curvature identities hold on normal lc $ ACy $ -manifolds, while the contact analog of A. Gray’s first identity holds if and only if the manifold is cosymplectic.     

Symbolic extensions and dominated splittings for generic C^{1}-diffeomorphisms

Let $\mathrm{Diff }^1(M)$ be the set of all $C^1$ -diffeomorphisms $f:M\rightarrow M$ , where $M$ is a compact boundaryless d-dimensional manifold, $d\ge 2$ . We prove that there is a residual subset $\mathfrak R $ of $\mathrm{Diff }^1(M)$ such that if $f\in \mathfrak R $ and if $H(p)$ is the homoclinic class associated with a hyperbolic periodic point $p$ , then either $H(p)$ admits a dominated splitting of the form $E\oplus F_1\oplus \dots \oplus F_k\oplus G$ , where $F_i$ is not hyperbolic and one-dimensional, or $f|_{H(p)}$ has no symbolic extensions.     

The polynomial dual of an operator ideal

We prove that the adjoint of a continuous homogeneous polynomial $P$ between Banach spaces belongs to a given operator ideal $\mathcal I$ if and only if $P$ admits a factorization $P = u \circ Q$ where the adjoint of the linear operator $u$ belongs to $\mathcal I$ . Several consequences of this factorization are obtained, for example we characterize the polynomials whose adjoints are absolutely $p$ -summing.     

Free subalgebras of division algebras over uncountable fields

We study the existence of free subalgebras in division algebras, and prove the following general result: if $A$ is a noetherian domain which is countably generated over an uncountable algebraically closed field $k$ of characteristic $0$ , then either the quotient division algebra of $A$ contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if $k$ is an uncountable algebraically closed field and $A$ is a finitely generated $k$ -algebra that is a domain of GK-dimension strictly less than $3$ , then either $A$ satisfies a polynomial identity, or the quotient division algebra of $A$ contains a free $k$ -algebra on two generators.     

Caffarelli–Kohn–Nirenberg inequality on metric measure spaces with applications

We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli–Kohn–Nirenberg inequality with the same exponent $n \ge 3$ , then it has exactly the $n$ -dimensional volume growth. As an application, if an $n$ -dimensional Finsler manifold of non-negative $n$ -Ricci curvature satisfies the Caffarelli–Kohn–Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.     

Rigidity of Einstein manifolds with positive scalar curvature

We prove that if $M^n(n\ge 4)$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $R_0>\sigma _{n}K_{\max }$ , where $\sigma _n\in (\frac{1}{4},1)$ is an explicit positive constant depending only on $n$ , then $M$ must be isometric to a spherical space form. Moreover, we prove that if an $n(\ge {\!\!4})$ -dimensional compact Einstein manifold satisfies $K_{\min }\ge \eta _n R_0,$ where $\eta _n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is locally symmetric. It should be emphasized that the pinching constant $\eta _n$ is optimal when $n$ is even. We then obtain some rigidity theorems for Einstein manifolds under $(n-2)$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $M$ is an $n(\ge {\!\!4})$ -dimensional compact Einstein submanifold in the simply connected space form $F^{N}(c)$ with constant curvature $c\ge 0$ , and the normalized scalar curvature $R_0$ of $M$ satisfies $R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$ where $A_n=n^3-5n^2+8n$ , and $H$ is the mean curvature of $M$ , then $M$ is isometric to a standard $n$ -sphere.     

Duality and the topological filtration

Let $(M,J)$ be a Fano manifold which admits a Kähler-Einstein metric $g_{KE}$ (or a Kähler-Ricci soliton $g_{KS}$ ). Then we prove that Kähler-Ricci flow on $(M,J)$ converges to $g_{KE}$ (or $g_{KS}$ ) in $C^\infty $ in the sense of Kähler potentials modulo holomorphisms transformation as long as an initial Kähler metric of flow is very closed to $g_{KE}$ (or $g_{KS}$ ). The result improves Main Theorem in [14] in the sense of stability of Kähler-Ricci flow.     

On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems

We prove the existence of infinitely many homoclinic orbits on a Riemannian manifold (possibly non-compact), for a class of second order Hamiltonian systems of the form: $$D_t \dot x(t) + grad_x V(t,x(t)) = 0$$ where the potentialV isT-periodic in the time variable.