Existence Of Foliation Research Articles

A-foliations of codimension two on compact simply-connected manifolds

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian (n+2)-manifold, with regular leaves homeomorphic to the n-torus, is given by a smooth effective n-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds.

On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Given an exterior domainΩ\OmegawithC2,αC^{2,\alpha }boundary inRn\mathbb {R}^{n},n≥3n\geq 3, we obtain a11-parameter familyuγ∈C∞(Ω)u_{\gamma }\in C^{\infty }\left ( \Omega \right ),|γ|≤π/2\left \vert \gamma \right \vert \leq \pi /2, of solutions of the minimal surface equation such that, if|γ|>π/2\left \vert \gamma \right \vert >\pi /2,uγ∈C∞(Ω)∩C2,α(Ω¯)u_{\gamma }\in C^{\infty }\left ( \Omega \right ) \cap C^{2,\alpha }\left ( \overline {\Omega }\right ),uγ|∂Ω=0u_{\gamma }|_{\partial \Omega }=0withmax∂Ω‖∇uγ‖=tan⁡γ\max _{\partial \Omega }\left \Vert \nabla u_{\gamma }\right \Vert =\tan \gammaand, if|γ|=π/2\left \vert \gamma \right \vert =\pi /2, the graph ofuγu_{\gamma }is contained in aC1,1C^{1,1}manifoldMγ⊂Ω¯×RM_{\gamma }\subset \overline {\Omega }\times \mathbb {R}with∂Mγ=∂Ω\partial M_{\gamma }=\partial \Omega. Each of these functions is bounded and asymptotic to a constant\[cγ=lim‖x‖→∞uγ(x).c_{\gamma }=\lim _{\left \Vert x\right \Vert \rightarrow \infty }u_{\gamma }\left ( x\right ) .\]The mappingsγ→uγ(x)\gamma \rightarrow u_{\gamma }\left ( x\right )(for fixedx∈Ωx\in \Omega) andγ→cγ\gamma \rightarrow c_{\gamma }are strictly increasing and bounded. The graphs of these functions foliate the open subset ofRn+1\mathbb {R}^{n+1}\[{(x,z)∈Ω×R, −uπ/2(x)>z>uπ/2(x)}.\left \{ \left ( x,z\right ) \in \Omega \times \mathbb {R}\text {, }-u_{\pi /2}\left ( x\right ) >z>u_{\pi /2}\left ( x\right ) \right \} .\]Moreover, ifRn∖Ω\mathbb {R}^{n}\backslash \Omegasatisfies the interior sphere condition of maximal radiusρ\rhoand if∂Ω\partial \Omegais contained in a ball of minimal radiusϱ\varrho, then\[[0,σnρ]⊂[0,cπ/2]⊂[0,σnϱ],\left [ 0,\sigma _{n}\rho \right ] \subset \left [ 0,c_{\pi /2}\right ] \subset \left [ 0,\sigma _{n}\varrho \right ] ,\]where\[σn=∫1∞dtt2(n−1)−1.\sigma _{n}=\int _{1}^{\infty }\frac {dt}{\sqrt {t^{2\left ( n-1\right ) }-1}}.\]One of the above inclusions is an equality if and only ifρ=ϱ\rho =\varrho,Ω\Omegais the exterior of a ball of radiusρ\rhoand the solutions are radial.These foliations were studied by E. Kuwert in [Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), pp. 445–451] and our result answers a natural question about the existence of such foliations which was not touched by Kuwert.

Compact leaves of codimension one holomorphic foliations on projective manifolds

This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.

Existence of foliations on 4–manifolds

We present existence results for certain singular 2-dimensional foliations on 4-manifolds. The singularities can be chosen to be simple, e.g. the same as those that appear in Lefschetz pencils. There seems to be a wealth of such creatures on most 4-manifolds. In certain cases, one can prescribe surfaces to be transverse or be leaves of these foliations. The purpose of this paper is to offer objects, hoping for a future theory to be developed on them. For example, foliations that are taut might offer genus bounds for embedded surfaces (Kronheimer's conjecture).

The volume expansion rate and the age of the Universe

Under four assumptions such that 1) Einstein's theory of gravity is correct, 2) existence of foliation by geodesic slicing, 3) the trace of the extrinsic curvature, $K$, is negative at the present time in observed region, that is, the observed universe is now expanding, 4) the strong energy condition is satisfied, we show that $H_{v0} t_{0} \leq 1$, where $H_v \equiv - K/3$ agrees with the Hubble parameter in the case of a homogeneous and isotropic universe, and $t_0$ is the age of the Universe. If $H_{v0} t_0 > 1$ is confirmed observationally, at least one of the four assumptions is incorrect.