Showing papers in "Mathematische Annalen in 2007"

Independence in topological and C*-dynamics

Abstract: We develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods. One of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties. We also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to C*-dynamics.

On the long-time behavior of type-III Ricci flow solutions

Abstract: We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton’s compactness theorem that does not assume a lower injectivity radius bound, in terms of Riemannian groupoids. Using this, we show that the long-time behavior of type-III Ricci flow solutions is governed by the dynamics of an \({\mathbb{R}}^+\) -action on a compact space.

Infinite Dimensional Duality and Applications

Abstract: The usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear. In this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi-relative interior, which, in all the concrete problems considered, is nonempty. We apply this theory to solve the until now unsolved problem of finding, in the infinite dimensional case, the Lagrange multipliers associated to optimization problems or to variational inequalities. As an example, we find the Lagrange multiplier associated to a general elastic–plastic torsion problem.

On well-posedness for the Benjamin-Ono equation

Abstract: We prove existence and uniqueness of solutions for the Benjamin–Ono equation with data in \(H^{s}({\mathbb{R}})\) , s > 1/4. Moreover, the flow is holder continuous in weaker topologies.

Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series

Abstract: Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salie sums, and a well known reformulation of Salie sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincare series in Kohnen’s Γ0(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.

Local behaviour of solutions to doubly nonlinear parabolic equations

Abstract: We give a relatively simple and transparent proof for Harnack’s inequality for certain degenerate doubly nonlinear parabolic equations. We consider the case where the Lebesgue measure is replaced with a doubling Borel measure which supports a Poincare inequality.

\({\mathcal{C}}_{0}\) (X)-algebras, stability and strongly self-absorbing \({\mathcal{C}}^{*}\) -algebras

Abstract: We study permanence properties of the classes of stable and so-called \({\mathcal{D}}\)-stable \({\mathcal{C}}^{*}\)-algebras, respectively. More precisely, we show that a \({\mathcal{C}}_{0}\) (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for \({\mathcal{C}}^{*}\)-algebras absorbing the Jiang–Su algebra \({\mathcal{Z}}\) tensorially). Furthermore, we prove that if \({\mathcal{D}}\) is a K 1-injective strongly self-absorbing \({\mathcal{C}}^{*}\)-algebra, then A absorbs \({\mathcal{D}}\) tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a \({\mathcal{C}}^{*}\)-algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of \({\mathcal{Z}}\)-absorbing \({\mathcal{C}}^{*}\) -algebras.

Existence of analytic solutions for the classical Stefan problem

Abstract: We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a (unique) solution that is analytic in space and time.

Regularity results for quasilinear elliptic equations in the Heisenberg group

Abstract: We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg group \({\mathbb{H}}^n\) . The model case is the non-degenerate p-Laplacean operator \(\sum_{i=1}^{2n} X_i \left( \left(\mu^2+ \left| {\mathfrak{X}}u \right|^2\right)^\frac{p-2}{2} X_i u\right) =0,\) where \(\mu > 0\) , and p is not too far from 2.

Eigenvalues and energy functionals with monotonicity formulae under Ricci flow

Abstract: In this note, we construct families of functionals of the type of $${\mathcal{F}}$$ -functional and $${\mathcal{W}}$$ -functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a proof of the theorem that compact steady Ricci breathers must be Ricci-flat. Using these new functionals, we also give a new proof of Perelman’s no non-trivial expanding breather theorem. Furthermore, we prove that compact expanding Ricci breathers must be Einstein by a direct method. In this note, we also extend Cao’s methods of eigenvalues (in Math Ann 337(2), 2007) and improve their results.

On the Beauville form of the known irreducible symplectic varieties

Abstract: We study the global geometry of O’Grady’s ten-dimensional irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.

Recasting the Elliott conjecture

Abstract: Let A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang–Su algebra \({\mathcal{Z}}\) tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—\({\mathcal{Z}}\) -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of \({\mathcal{Z}}\) -stable C*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that \({\mathrm{K}}\) -theoretic invariants will classify separable and nuclear C*-algebras—with the recent appearance of counterexamples to its strongest concrete form.

On mod p properties of Siegel modular forms

Abstract: We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator \(\varTheta\) on q-expansions and show that the algebra of Siegel modular forms mod p is stable under \({\varTheta}\), by exploiting the relation between \({\varTheta}\) and generalized Rankin-Cohen brackets.

On a problem of K. Mahler: Diophantine approximation and Cantor sets

Abstract: Let K denote the middle third Cantor set and \(\mathcal{A}:= \{ 3^{n} : n = 0,1,2, \ldots \}\). Given a real, positive function ψ let \( W_{\mathcal{A}}(\psi)\) denote the set of real numbers x in the unit interval for which there exist infinitely many \((p,q) \in \mathbb{Z} \times {\mathcal{A}}\) such that |x − p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin–Schaeffer conjecture is established for \(W_{\mathcal{A}}(\psi) \cap K\). One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K—an assertion attributed to K. Mahler. Explicit examples of irrational numbers satisfying Mahler’s assertion are also given.

The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry

Abstract: Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths.

Galois cohomology and forms of algebras over Laurent polynomial rings

Abstract: 3 Loop torsors 8 3.1 The Witt-Tits index of a loop torsor . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Almost commutative subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Almost commuting families of elements of finite order . . . . . . . . . . . . 16 3.5 Almost commuting pairs and their invariants . . . . . . . . . . . . . . . . . 17 3.6 Failure in the anisotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

Abstract: The integrability of an m-component system of hydrodynamic type, u t = V(u)u x , by the generalized hodograph method requires the diagonalizability of the m × m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ × ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

Abstract: In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier–Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $$\dot{B}^0_{\infty, 1}(\mathbb{R}^3)$$ . For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L q (B(z,r)) for some $$q\in [2, \infty)$$ , where the ball of radius r is shrinking toward a possible singularity point z at the order of $$\sqrt{T-t}$$ as t approaches to T. In the $$L^q (\mathbb{R}^3)$$ convergence case with $$q\in [3, \infty)$$ we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126.

A relation between the parabolic Chern characters of the de Rham bundles

Abstract: In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism $${f:X_U\longrightarrow U}$$ for $${i\geq 0}$$ . We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.

Rate of Type II blowup for a semilinear heat equation

Abstract: A solution u of a Cauchy problem for a semilinear heat equation $$\left\{ \begin{array}{ll}u_{t} = \Delta u + u^{p} & \quad {\rm in}\, {\bf R}^N \times (0,\,T),\\u(x,0) = u_{0}(x) \geq 0 & \quad {\rm in}\, {\bf R}^N \end{array} \right.$$ is said to undergo Type II blowup at t = T if lim sup \(_{t earrow T} \; (T-t)^{1/(p-1)} |u(t)|_\infty = \infty .\) Let \(\varphi_\infty\) be the radially symmetric singular steady state. Suppose that \(u_0 \in L^\infty\) is a radially symmetric function such that \(u_0 - \varphi_\infty\) and (u0)t change sign at most finitely many times. We determine the exact blowup rate of Type II blowup solution with initial data u0 in the case of p > pL, where pL is the Lepin exponent.

Commuting elements and spaces of homomorphisms

Abstract: This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting.

Amenability, tubularity, and embeddings into \mathcal{R}^{\omega}

Abstract: Suppose M is a tracial von Neumann algebra embeddable into \(\mathcal{R}^{\omega}\) (the ultraproduct of the hyperfinite II1-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any e > 0 there exist \(m \in \mathbb{N}\) and γ > 0 such that if \((x_{1},\ldots, x_{n}), (y_{1}, \ldots, y_{n}) \in \Gamma(X;m,k,\gamma),\) then there exists a k × k unitary u satisfying \(|ux_iu^* - y_i|_2 < \epsilon\) for each 1 ≤ i ≤ n. We show that the following conditions are equivalent: M is amenable (i.e., injective). X is tubular. Any two embeddings of M into \(\mathcal{R}^{\omega}\) are conjugate by a unitary \(u \in \mathcal {R}^{\omega}\).

Sharp upper bounds for a variational problem with singular perturbation

Abstract: Let Ω be a C2 bounded open set of \(\mathbb{R}^{2}\) and consider the functionals \(F_{\epsilon} (u) := \int\limits_{\Omega} \left\{\frac{(1-|{ abla} u (x)|^{2})^{2}}{\epsilon} + {\epsilon} |D^{2} u (x)|^{2}\right\} {\rm d}x\) We prove that if \(u\in W^{1, \infty} (\Omega)\), |∇ u| = 1 a.e., and ∇ u∈BV, then \(\Gamma-\lim\limits_{\epsilon\downarrow0} F_\epsilon (u)=\frac13 \int\limits_{J_{ abla u}} |[ abla u]|^{3} {\rm d}\fancyscript{H}^{1}.\) The new result is the Γ- lim sup inequality.

Higher cohomology of divisors on a projective variety

Abstract: If L is ample, or merely nef, then of course hi ( X,OX(mL) ) = o(md) for i > 0. In general the converse is false: for instance, it can happen that H i ( X,OX(mL) ) = 0 for m > 0 and all i even if L is not nef (or, for that matter, pseudoeffective). However our main result shows that if one considers also small perturbations of the divisor in question, then in fact the maximal growth of higher cohomology characterizes non-ample divisors:

Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation

Abstract: We establish new lower bounds for the number of nodal bound states for the semiclassical nonlinear Schrodinger equation $$-\varepsilon^2 \Delta u+ a(x)u=|u|^{p-2}u$$ with bounded and uniformly continuous potential a. The solutions we obtain have precisely two nodal domains, and their positive and negative parts concentrate near the set of minimum points of a. Our approach is independent of penalization techniques and yields, in some cases, the existence of infinitely many nodal solutions for fixed $$\varepsilon$$ . Via a dynamical systems approach, we exhibit positively invariant sets of sign changing functions for the negative gradient flow of the associated energy functional. We analyze these sets on the cohomology level with the help of Dold’s fixed point transfer. In particular, we estimate their cuplength in terms of the cuplength of equivariant configuration spaces of subsets of $$\mathbb{R}^N$$ . We also provide new estimates of the cuplength of configuration spaces.

Extensions of linking systems with p-group kernel

Abstract: We study extensions of p-local finite groups where the kernel is a p-group. In particular, we construct examples of saturated fusion systems \({\mathcal{F}}\) which do not come from finite groups, but which have normal p-subgroups \({A \vartriangleleft \mathcal{F}}\) such that \({\mathcal{F}/A}\) is the fusion system of a finite group. One of the tools used to do this is the concept of a “transporter system”, which is modelled on the transporter category of a finite group, and is more general than a linking system.

Maximum principle for fully nonlinear equations via the iterated comparison function method

Abstract: We present various versions of generalized Aleksandrov–Bakelman–Pucci (ABP) maximum principle for L p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the “iterated” comparison function method, which was introduced in our previous paper (Koike and Świech in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Świech in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5–6), 967–983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293–1326, 1998; Comm. Partial Diff. Eq. 25, 1997–2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27–76, 1992) and (Crandall and Świech in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case.

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

Abstract: We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples.

The Stokes operator in weighted L q -spaces II: weighted resolvent estimates and maximal L p -regularity

Abstract: In this paper we establish a general weighted L q -theory of the Stokes operator $${\mathcal{A}}_{q,\omega}$$ in the whole space, the half space and a bounded domain for general Muckenhoupt weights $$\omega \in A_q$$ . We show weighted L q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator $${\mathcal{A}}_{q,\omega}$$ generates a bounded analytic semigroup but even yield the maximal L p -regularity of $${\mathcal{A}}_{q,\omega}$$ in the respective weighted L q -spaces for arbitrary Muckenhoupt weights $$\omega \in A_q$$ . This conclusion is archived by combining a recent characterisation of maximal L p -regularity by $${\mathcal{R}}$$ -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L p -regularity. Preprint (1999)] with the fact that for L q -spaces $${\mathcal{R}}$$ -boundedness is implied by weighted estimates.

L^2-Betti numbers of one-relator groups

Abstract: We determine the L 2 -Betti numbers of all one-relator groups and all surface-plus-one-relation groups. We also obtain some information about the L 2 -cohomology of left-orderable groups, and deduce the non-L 2 result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free.