### Applications of convex integration to symplectic and contact geometry

We apply Gromov’s method of convex integration to problems related to the existence and uniqueness of symplectic and contact structures

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We apply Gromov’s method of convex integration to problems related to the existence and uniqueness of symplectic and contact structures

The introduction of the concepts of energy machinery and energy structure on a manifold makes it possible a large class of energy representations of gauge groups including, as a very particular case, the ones known up to now. By using an adaptation of methods initiated by I. M. Gelfand, we provide a sufficient condition for the irreducibility of these representations.

We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...

The $k$-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations ${F}_{k}\left[u\right]=0$, where ${F}_{k}\left[u\right]$ is the elementary symmetric function of order $k$, $1\le k\le n$, of the eigenvalues of the Hessian matrix ${D}^{2}u$. For example, ${F}_{1}\left[u\right]$ is the Laplacian $\Delta u$ and ${F}_{n}\left[u\right]$ is the real Monge-Ampère operator det ${D}^{2}u$, while $1$-convex functions and $n$-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative $k$-convex functions, and give several...

We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.

Let ${\Phi}_{1},...,{\Phi}_{k+1}$ (with $k\ge 1$) be vector fields of class ${C}^{k}$ in an open set $U{\subset}^{N+m}$, let $\mathbb{M}$ be a $N$-dimensional ${C}^{k}$ submanifold of $U$ and define $$\mathbb{T}:=\{z\in \mathbb{M}:{\Phi}_{1}\left(z\right),...,{\Phi}_{k+1}\left(z\right)\in {T}_{z}\mathbb{M}\}$$ where ${T}_{z}\mathbb{M}$ is the tangent space to $\mathbb{M}$ at $z$. Then we expect the following property, which is obvious in the special case when ${z}_{0}$ is an interior point (relative to $\mathbb{M}$) of $\mathbb{T}$: If ${z}_{0}\in \mathbb{M}$ is a $(N+k)$-density point (relative to $\mathbb{M}$) of $\mathbb{T}$ then all the iterated Lie brackets of order less or equal to $k$$$...$$

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure $\mu \left(x\right)=N{\sum}_{k=1}^{N}{x}_{k}$ μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is...

Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and ${\mathscr{H}}_{E}(M,\omega )$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism ${c}^{\omega}:{\mathscr{H}}_{E}(M,\omega )\to (M,\omega )$, which measures for each $h\in {\mathscr{H}}_{E}(M,\omega )$ the mass flow toward ends induced by h. We show that the map ${c}^{\omega}$ has...

The notion of an implicit Hamiltonian system-an isotropic mapping H: M → (TM,ω̇) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated.