- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 20:58:50
6
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=7&iss=1&update_since=2024-03-28
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2005
1
On real Kähler Euclidean submanifolds with non-negative Ricci curvature
Luis
Florit
Estrada Dona Castorina 110, RIO DE JANEIRO RJ, BRAZIL
Wing San
Hui
Ohio State University, COLUMBUS, UNITED STATES
F.
Zheng
Ohio State University, COLUMBUS, UNITED STATES
Kähler submanifolds, Ricci curvature, holomorphic curvature
We show that any real K\"ahler Euclidean submanifold \fk with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to $2n-2p$. Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that $M^{2n}$ is complete. In particular, we conclude that the only real K\"ahler submanifolds $M^{2n}$ in $\R^{3n}$ that have either positive Ricci curvature or positive holomorphic sectional curvature are precisely products of $n$ orientable surfaces in $\R^3$ with positive Gaussian curvature. Further applications of our main result are also given.
Differential geometry
General
1
11
10.4171/JEMS/19
http://www.ems-ph.org/doi/10.4171/JEMS/19
Relative maps and tautological classes
Carel
Faber
Royal Institute of Technology, STOCKHOLM, SWEDEN
Rahul
Pandharipande
ETH Zürich, ZÜRICH, SWITZERLAND
Global analysis, analysis on manifolds
General
13
49
10.4171/JEMS/20
http://www.ems-ph.org/doi/10.4171/JEMS/20
Measures of maximal entropy for random $\beta$-expansions
Karma
Dajani
Universiteit Utrecht, UTRECHT, NETHERLANDS
Martijn
de Vries
Vrije Universiteit, AMSTERDAM, NETHERLANDS
greedy expansions, lazy expansions, Markov chains, measures of maximal entropy
Let $\beta >1$ be a non-integer. We consider $\beta$-expansions of the form $\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]$. We show that $K_{\beta}$ has a unique mixing measure $\nu_{\beta}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure $\nu_{\beta}$ the digits $(d_i)_{i \geq 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of $\beta$-expansions.
Measure and integration
General
51
68
10.4171/JEMS/21
http://www.ems-ph.org/doi/10.4171/JEMS/21
Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues
Anton
Bovier
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Véronique
Gayrard
CNRS Luminy, MARSEILLE CEDEX 9, FRANCE
Markus
Klein
Universität Potsdam, POTSDAM, GERMANY
Metastability, diffusion processes, spectral theory, potential theory, capacity, exit timespotential theory, capacity, exit times
We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in \cite{BEGK3}, with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form $-\e \Delta +\nabla F(\cdot)\nabla$ on $\R^d$ or subsets of $\R^d$, where $F$ is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius $\e$ centered at the positions of the local minima of $F$. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In \cite{BEGK3} it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical {\it Eyring-Kramers formula}.
Statistical mechanics, structure of matter
Probability theory and stochastic processes
General
69
99
10.4171/JEMS/22
http://www.ems-ph.org/doi/10.4171/JEMS/22
Coxeter group actions on the complement of hyperplanes and special involutions
Giovanni
Felder
ETH Zürich, ZÜRICH, SWITZERLAND
Alexander
Veselov
Loughborough University, LOUGHBOROUGH, UNITED KINGDOM
Coxeter groups, hyperplane arrangements, Brieskorn's braid groups
We consider both standard and twisted action of a (real) Coxeter group $G$ on the complement $\mathcal M_G$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^*(\mathcal M_G, {\mathbb C})$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_n$, the Weyl groups of type $D_{2m+1}$, $E_6$ and dihedral groups $I_2 (2k+1).$ We discuss also the relations with the cohomology of Brieskorn's braid groups.
Topological groups, Lie groups
General
101
116
10.4171/JEMS/23
http://www.ems-ph.org/doi/10.4171/JEMS/23
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
Antonio
Ambrosetti
SISSA, TRIESTE, ITALY
Veronica
Felli
Università degli Studi di Milano-Bicocca, MILANO, ITALY
Andrea
Malchiodi
Scuola Normale Superiore, PISA, ITALY
Nonlinear Schrödinger equations, weighted Sobolev spaces
We deal with a class on nonlinear Schr\"odinger equations \eqref{eq:1} with potentials $V(x)\sim |x|^{-\a}$, $0
Partial differential equations
General
117
144
10.4171/JEMS/24
http://www.ems-ph.org/doi/10.4171/JEMS/24