Gaussian bounds for derivatives of central Gaussian semigroups on compact groups
A.
Bendikov;
L.
Saloff-Coste
1279-1298
Abstract: For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumptions, we prove that these derivatives satisfy certain Gaussian bounds.
An analogue of minimal surface theory in $\operatorname{SL}(n,\mathbf C)/\operatorname{SU}(n)$
M.
Kokubu;
M.
Takahashi;
M.
Umehara;
K.
Yamada
1299-1325
Abstract: We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. $\operatorname{SL}(n,\mathbf{C})/\operatorname{SU}(n)$), which contains minimal surfaces in $\mathbf{R}^n$ and constant mean curvature $1$ surfaces in $\mathcal{H}^3$. A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.
Determinacy and weakly Ramsey sets in Banach spaces
Joan
Bagaria;
Jordi
López-Abad
1327-1349
Abstract: We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of $c_0$ we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper ``Weakly Ramsey sets in Banach spaces.''
Milnor classes of local complete intersections
J.-P.
Brasselet;
D.
Lehmann;
J.
Seade;
T.
Suwa
1351-1371
Abstract: Let $V$ be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component $S$ of the singular set $\operatorname{Sing}(V)$ of $V$, we define the Milnor class $\mu _{*}(V,S)$ in the homology of $S$. The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of $V$ is shown to be equal to the sum of $\mu _{*}(V,S)$ over the connected components $S$ of $\operatorname{Sing}(V)$. This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The $0$-degree component $\mu _{0}(V,S)$ coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for $\mu _{*}(V,S)$ when $S$ is a non-singular component and $V$ satisfies the Whitney condition along $S$.
Dual decompositions of 4-manifolds
Frank
Quinn
1373-1392
Abstract: This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index $\leq 2$. In dimensions $\geq 5$results of Smale (trivial $\pi _{1}$) and Wall (general $\pi _{1}$) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ( $H_{1}K\simeq H^{2}L$ and $H_{2}K\simeq H^{1}L$). (3.3) If $(N,\partial N)$ is 1-connected then there is a ``pseudo'' handle decomposition without 1-handles, in the sense that there is a pseudo collar $(M,\partial N)$ (a relative 2-handlebody with spine that 2-deforms to $\partial N$) and $N$ is obtained from this by attaching handles of index $\geq 2$.
Second order Lagrangian Twist systems: simple closed characteristics
J.
B.
Van den Berg;
R.
C.
Vandervorst
1393-1420
Abstract: We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is defined via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels. This reduction has some similarities with the method of broken geodesics for the geodesic variational problem on Riemannian manifolds. On the other hand, the monotonicity property can be related to the existence of local Twist maps in the associated Hamiltonian flow. The finite dimensional reduction gives rise to a second order monotone recurrence relation. We study these recurrence relations to find simple closed characteristics for the Lagrangian system. More complicated closed characteristics will be dealt with in future work. Furthermore, we give conditions on the Lagrangian that guarantee the Twist property.
Global existence for a quasi-linear evolution equation with a non-convex energy
Eduard
Feireisl;
Hana
Petzeltová
1421-1434
Abstract: We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.
Weak amenability of triangular Banach algebras
B.
E.
Forrest;
L.
W.
Marcoux
1435-1452
Abstract: Let ${\mathcal A}$ and ${\mathcal B}$be unital Banach algebras, and let ${\mathcal M}$ be a Banach ${\mathcal A},{\mathcal B}$-module. Then ${\mathcal T} = \left[ \begin{array}{cc} {\mathcal A} & {\mathcal M} 0 & {\mathcal B} \end{array} \right]$ becomes a triangular Banach algebra when equipped with the Banach space norm $\ensuremath {\Vert}\left[ \begin{array}{cc} a & m 0 & b \end{array} \right] \... ...rt} _{{\mathcal M}} + \ensuremath {\Vert} b \ensuremath {\Vert} _{{\mathcal B}}$. A Banach algebra ${\mathcal T}$is said to be $n$-weakly amenable if all derivations from ${\mathcal T}$ into its $n^{\mathrm{th}}$ dual space ${\mathcal T}^{(n)}$ are inner. In this paper we investigate Arens regularity and $n$-weak amenability of a triangular Banach algebra ${\mathcal T}$ in relation to that of the algebras ${\mathcal A}$, ${\mathcal B}$ and their action on the module ${\mathcal M}$.
SRB measures and Pesin's entropy formula for endomorphisms
Min
Qian;
Shu
Zhu
1453-1471
Abstract: We present a formulation of the SRB (Sinai-Ruelle-Bowen) property for invariant measures of $C^2$ endomorphisms (maybe non-invertible and with singularities) of a compact manifold via their inverse limit spaces, and prove that this property is necessary and sufficient for Pesin's entropy formula. This result is a non-invertible endomorphisms version of a result of Ledrappier, Strelcyn and Young.
Wandering orbit portraits
Jan
Kiwi
1473-1485
Abstract: We study a counting problem in holomorphic dynamics related to external rays of complex polynomials. We give upper bounds on the number of external rays that land at a point $z$ in the Julia set of a polynomial, provided that $z$has an infinite forward orbit.
Product systems over right-angled Artin semigroups
Neal
J.
Fowler;
Aidan
Sims
1487-1509
Abstract: We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid $\mathcal G$. We show that the existing notions of product systems fit into our categorical framework, as do the $k$-graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid $\mathcal G$. In particular, we obtain necessary and sufficient conditions under which a collection of $k$ $1$-graphs form the coordinate graphs of a $k$-graph.
Tractor calculi for parabolic geometries
Andreas
Cap;
A.
Rod
Gover
1511-1548
Abstract: Parabolic geometries may be considered as curved analogues of the homogeneous spaces $G/P$ where $G$ is a semisimple Lie group and $P\subset G$ a parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.
Block representation type of reduced enveloping algebras
Iain
Gordon;
Alexander
Premet
1549-1581
Abstract: Let $K$ be an algebraically closed field of characteristic $p$, $G$ a connected, reductive $K$-group, $\mathfrak{g}=\text{Lie}(G)$, $\chi\in\mathfrak{g}^*$ and $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. Assume that $G^{(1)}$ is simply-connected, $p$ is good for $G$ and $\mathfrak{g}$ has a non-degenerate $G$-invariant bilinear form. All blocks of $U_\chi(\mathfrak{g})$ having finite and tame representation type are determined.
Nonradial solvability structure of super-diffusive nonlinear parabolic equations
Panagiota
Daskalopoulos;
Manuel
del Pino
1583-1599
Abstract: We study the solvability of the Cauchy problem for the nonlinear parabolic equation \begin{displaymath}\frac {\partial u}{\partial t} = \mbox{div}\, (u^{m-1}\nabla u)\end{displaymath} when $m < 0$ in ${\bf R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=\vert x\vert$ for the spherical averages of $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= \vert x\vert$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ vanishes on some interval of length $l^* = \frac {(m-1)\pi}{2m} >0$, then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.
The index of a critical point for densely defined operators of type $(S_+)_L$ in Banach spaces
Athanassios
G.
Kartsatos;
Igor
V.
Skrypnik
1601-1630
Abstract: The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type $(S_{+})_{L}$ acting from a real, reflexive and separable Banach space $X$ into $X^{*}.$ This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in $l^{p},~p>2,$ illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type $(S_{+})).$ Applications to nonlinear Dirichlet problems have appeared elsewhere.
Extremal problems for quasiconformal maps of punctured plane domains
Vladimir
Markovic
1631-1650
Abstract: The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.
Convergence of two-dimensional weighted integrals
Malabika
Pramanik
1651-1665
Abstract: A two-dimensional weighted integral in $\mathbb R^{2}$ is proposed as a tool for analyzing higher-dimensional unweighted integrals, and a necessary and sufficient condition for the finiteness of the weighted integral is obtained.
Monge's transport problem on a Riemannian manifold
Mikhail
Feldman;
Robert
J.
McCann
1667-1697
Abstract: Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures $\mu^+ \ne \mu^-$, find the measure-preserving map $s:M \longrightarrow M$ between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold $M$ -- and assuming absolute continuity of $\mu^+$ -- an optimal map will be shown to exist. Aspects of its uniqueness are also established.
An estimate for weighted Hilbert transform via square functions
S.
Petermichl;
S.
Pott
1699-1703
Abstract: We show that the norm of the Hilbert transform as an operator on the weighted space $L^2(w)$ is bounded by a constant multiple of the $3/2$ power of the $A_2$ constant of $w$, in other words by $c\, \sup_I (\langle \omega \rangle_I \langle \omega^{-1} \rangle_I)^{3/2}$. We also give a short proof for sharp upper and lower bounds for the dyadic square function.