Geometric and Functional Analysis最新文献

We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3–manifolds and answer questions of Agol–Liu (J. Am. Math. Soc. 25(1):151–187, 2012) and Reid–Wang–Zhou (Acta Math. Sin. Engl. Ser. 18(1):157–172, 2002).

By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Guéritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Let (M,g₀) be a closed oriented hyperbolic manifold of dimension at least 3. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric g on M with same volume as g₀, its volume entropy h(g) satisfies h(g)≥n−1 with equality only when g is isometric to g₀. We show that the hyperbolic metric g₀ is stable in the following sense: if g_i is a sequence of Riemaniann metrics on M of same volume as g₀ and if h(g_i) converges to n−1, then there are smooth subsets Z_i⊂M such that both (operatorname{Vol}(Z_{i},g_{i})) and (operatorname{Area}(partial Z_{i},g_{i})) tend to 0, and (M∖Z_i,g_i) converges to (M,g₀) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for M is intrinsically isomorphic to ((M,frac{(n-1)^{2}}{4n} g_{0})).

The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They admit a natural group action, and “random Brody curves” in the title refers to invariant probability measures for it. We study their geometric and dynamical properties. Given an invariant probability measure μ on the space of Brody curves, our first main theorem claims that its rate distortion dimension is bounded by the integral of a “geometric potential” over μ. This result is analogous to the Ruelle inequality of smooth ergodic theory. Our second main theorem claims that there exists a rich variety of invariant probability measures attaining equality in this “Ruelle inequality for Brody curves”. The main tools of the proofs are the deformation theory of Brody curves and the variational principle for mean dimension with potential. This approach is motivated by the theory of thermodynamic formalism for Axiom A diffeomorphisms.

Let G be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups satisfying the strong Atiyah conjecture over (K subseteq mathbb{C}) a field closed under complex conjugation. Assume that the orders of finite subgroups of G are bounded above. We show that G satisfies the strong Atiyah conjecture over K. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the ∗-regular closure of K[G] in (mathcal{U}(G)), (mathcal{R}_{K[G]}), is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding ∗-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over K are also closed under the graph of groups construction as long as the edge groups are finite. We also infer some consequences on the structure of the K₀ and K₁-groups of (mathcal{R}_{K[G]}). The techniques developed enable us to prove that K[G] fulfills the strong, algebraic and center-valued Atiyah conjectures, and that (mathcal{R}_{K[G]}) is the universal localization of K[G] over the set of all matrices that become invertible in (mathcal{U}(G)), provided that G belongs to a certain class of groups (mathcal{T}_{mathcal{VLI}}), which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.

We give an example of a family of smooth complex algebraic surfaces of degree 6 in (mathbb{CP}^{3}) developing an isolated elliptic singularity. We show via a gluing construction that the unique Kähler-Einstein metrics of Ricci curvature −1 on these sextics develop a complex hyperbolic cusp in the limit, and that near the tip of the forming cusp a Tian-Yau gravitational instanton bubbles off.

A rational lemniscate is a level set of |r| where (r: widehat {mathbb{C}}rightarrow widehat {mathbb{C}}) is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert’s lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge’s theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.

Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs. In this paper, we consider the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy’s law for incompressible porous media via buoyancy force. We prove that in contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions.

In this paper, we prove that an ancient smooth curve-shortening flow with finite entropy embedded in (mathbb{R}^{2}) has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplicity m≥3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.