Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

A density problem for Sobolev spaces on Gromov hyperbolic domains

We prove that for a bounded domain $Ω\subset \mathbb R^n$ which is Gromov hyperbolic with respect to the quasihyperbolic metric, especially when $Ω$ is a finitely connected planar domain, the Sobolev space $W^{1,\,\infty}(Ω)$ is dense in $W^{1,\,p}(Ω)$ for any $1\le p<\infty$. Moreover if $Ω$ is also Jordan or quasiconvex, then $C^{\infty}(\mathbb R^n)$ is dense in $W^{1,\,p}(Ω)$ for $1\le p<\infty$.

A Universal Lipschitz Extension Property of Gromov Hyperbolic Spaces

A metric space has the universal Lipschitz extension property if for each subspace S embedded quasi-isometrically into an arbitrary metric space M there exists a continuous linear extension of Banach-valued Lipschitz functions on S to those on all of M. We show that the finite direct sum of Gromov hyperbolic spaces of bounded geometry is universal in the sense of this definition.

Subelliptic estimates from Gromov hyperbolicity

In this paper we prove: if the complete Kähler-Einstein metric on a bounded convex domain (with no boundary regularity assumptions) is Gromov hyperbolic, then the $\bar{\partial}$-Neumann problem satisfies a subelliptic estimate. This is accomplished by constructing bounded plurisubharmonic function whose Hessian grows at a certain rate (which implies a subelliptic estimate by work of Catlin and Straube). We also provide a characterization of Gromov hyperbolicity in terms of orbit of the domain under the group of affine transformations. This characterization allows us to construct many example…

Banach Zuk's criterion for partite complexes with application to random groups

We prove a Banach version of Żuk's criterion for groups acting on partite simplicial complexes. Using this new criterion we derive a new fixed point theorem for random groups in the Gromov density model with respect to several classes of Banach spaces ($L^p$ spaces, Hilbertian spaces, uniformly curved spaces). In particular, we show that for every $p$, a group in the Gromov density model has asymptotically almost surely property $(F L^p)$ and give a sharp lower bound for the growth of the conformal dimension of the boundary of such group as a function of the parameters of the density model.

巴拿赫等距猜想的一个解

A solution to Banach's isometric conjecture

该研究解决了实巴拿赫空间的巴拿赫等距猜想，通过结合纤维丛拓扑与布劳威尔度理论，证明了所有奇数n的情形，与Gromov的偶数维结果共同完成了该猜想的实情形证明。

Wellposedness of hyperbolic evolution equations in Banach spaces

This paper has been withdrawn by the authors, because of a crucial gap in the proof of the main theorem.