Related papers: On the degenerated Arnold-Givental conjecture

Let (M,I, \omega, \Omega) be a nearly Kaehler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form \Omega and the Hermitian form \omega which satisfies $d\omega=3\lambda\Re\Omega, d\Im\Omega=-2\lambda\omega^2$, for a non-zero real…

We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel \mu…

In this paper we study variations of the Hopf theorem concerning continuous maps $f$ of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case when $M$ is a closed convex $n$-dimensional surface and…

We obtain some Liouville type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary and partially verifies Wang's conjecture (J. Geom. Anal. 31 (2021)). For…

By using the Yamabe flow, we prove that if $(M^n,g)$, $n\geq3$, is an $n$-dimensional locally conformally flat complete Riemannian manifold $Rc\geq \epsilon Rg>0$, where $\epsilon>0$ is a uniformly constant, then $M^n$ must be compact. Our…

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first…

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori…

Let $(M,\omega)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion…

We use sup-convolution to find upper approximations of a bounded $m$-subharmonic function on a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. As an application, we show the H\"older continuity of solutions to…

Given a Liouville manifold $M$, we introduce an invariant of $M$ that we call the Heegaard Floer symplectic cohomology $SH^*_\kappa(M)$ for any $\kappa \ge 1$ that coincides with the symplectic cohomology for $\kappa=1$. Writing $\hat{M}$…

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder\'on-Zygmund operators. Namely, given $1<p<q<\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight…

For a proper Hamiltonian action of a Lie group $G$ on a K\"ahler manifold $(X,\omega)$ with momentum map $\mu$ we show that the symplectic reduction $\mu^{-1}(0)/G$ is a normal complex space. Every point in $\mu^{-1}(0)$ has a $G$-stable…

In this note we give a short proof of Arnold's conjecture for the zero section of a cotangent bundle of a closed manifold. The proof is based on some basic properties of Lagrangian spectral invariants from Floer theory.

By solving the Cauchy problem for the Hodge-Laplace heat equation for $d$-closed, positive $(1, 1)$-forms, we prove an optimal gap theorem for K\"ahler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat…

Let a real-analytic manifold $M$ formally (holomorphically) equivalent to the following model…

We construct the vortex Floer homology group $VHF (M,\mu;H)$ for an aspherical Hamiltonian $G$-manifold $(M, \omega)$ with moment map $\mu$ and a class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3]. This is a…

Let $X$ be a Hamiltonian vector field defined on a symplectic manifold $(M,\omega)$, $g$ a nowhere vanishing smooth function defined on an open dense subset $M^0$ of $M$. We will say that the vector field $Y = gX$ is conformally…

For every compact almost complex manifold (M,J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to…

We deduce various norm equivalences, and convolution estimates for the modulation space $M^{\sharp ,q}_{(\omega )}$ consisting of all $f\in M^{\infty ,q}_{(\omega )}$ such that $|V_\phi f \cdot \omega |$ satisfies a mild vanishing condition…

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_\lambda \}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda =\lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension…