Equivariant Novikov Inequalities

We establish an equivariant generalization of the Novikov inequalities which allows us to estimate the topology of the set of critical points of a closed basic invariant form by means of twisted equivariant cohomology of the manifold. We apply these inequalities to study cohomology of the fixed points set of a symplectic torus action. We show that in this case our inequalities are perfect, i.e. they are in fact equalities.     

A remark on the Harnack inequality for non-self-adjoint evolution equations

In this paper we consider a non-self-adjoint evolution equation on a compact Riemannian manifold with boundary. We prove a Harnack inequality for a positive solution satisfying the Neumann boundary condition. In particular, the boundary of the manifold may be nonconvex and this gives a generalization to a theorem of Yau.

     

Morse–Bott inequalities in the presence of a compact Lie group action and applications

In this paper, we obtain Morse–Bott inequalities in the presence of a compact Lie group action via Bismut–Lebeauʼs analytic localization techniques. As an application, we obtain Morse–Bott inequalities on compact manifold with nonempty boundary by applying the generalized Morse–Bott inequalities to the doubling manifold.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

Fixed points of approximable maps

We present a simple proof of the Leray-Schauder type theorem for approximable multimaps given recently by Ben-El-Mechaiekh and Idzik. We apply this theorem to obtain a Schaefer type theorem, the Birkhoff-Kellogg type theorems, a Penot type theorem for non-self-maps, and quasi-variational inequalities, all related to compact closed approximable maps.

     

Notions of boundaries for function spaces

Sibony and the author independently defined a higher order generalization of the usual Shilov boundary of a function algebra which yielded extensions of results about analytic structure from one dimension to several dimensions. Tonev later obtained an alternative characterization of this generalized Shilov boundary by looking at closed subsets of the spectrum whose image under the spectral mapping contains the topological boundary of the joint spectrum. In this note we define two related notions of what it means to be a higher order/higher dimensional boundary for a space of functions without requiring that the boundary be a closed set. We look at the relationships between these two boundaries, and in the process we obtain an alternative proof of Tonev's result. We look at some examples, and we show how the same concepts apply to convex sets and linear functions.

     

Gradient estimates for the heat equation under the Ricci flow

The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the case where M is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on M.     

Porous media equation and Sobolev inequalities under negative curvature

We study the link between some modified porous media equation and Sobolev inequalities on a Riemannian manifold M whose Ricci curvature tensor is bounded below by a negative constant −ρ. The method used deals with entropy-energy differentiation and follows the way the author got inequalities under nonnegative Ricci curvature assumptions. The key of the proof is the curvature-dimension criterion.     

Strongly indefinite systems with critical Sobolev exponents

We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case.

     

A super-twisted Dirac operator and Novikov inequalities

A super-twisted Dirac operator is constructed and deformed suitably. Following Shubin’s approach to Novikov inequalities associated to the deformed de Rham-Hodge operator, we give a for mula for the index of the super-twisted Dirac operator, and Novikov type inequalities for the deformed operator. In particular, we obtain a purely analytic proof of the Hopf index theorem for general vector bundles.     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

In this article, we consider variational inequalities arising, e.g., in modelling diffusion of substances in porous media. We assume that the media fills a domain Ω_? of ? ⁿ with n?≥?3. We study the case where the number of cavities is large and they are periodically distributed along a (n???1)-dimensional manifold. ? is the period while ?^α is the size of each cavity with α?≥?1; ? is a parameter that converges towards zero. Moreover, we also assume that the nonlinear process involves a large parameter ?^?κ with κ?=?(α???1)(n???1). Passing to the scale limit, and depending on the value of α, the effective equation or variational inequality is obtained. In particular, we find a critical size of the cavities when α?=?κ?=?(n???1)/(n???2). We also construct correctors which improve convergence for α?≥?(n???1)/(n???2).     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

Inequalities for Functions with Higher Monotonicities

We consider real functions on [a, b] for which some derivatives have constant sign. For these functions we obtain Popoviciu and Favard-Berwald type inequalities as well as converse Holder inequalities.     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

Moment maps and Riemannian symmetric pairs

We study Hamiltonian actions of a compact Lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold. The fixed-point set of the involution on the manifold is a Lagrangian submanifold. We investigate its image under the moment map and conclude that the intersection with the Weyl chamber is an easily described subpolytope of the Kirwan polytope. Of special interest is the integral K?hler case, where much stronger results hold. In particular, we obtain convexity theorems for closures of orbits of the noncompact dual group (in the sense of the theory of symmetric pairs). In the abelian case these results were obtained earlier by Duistermaat. We derive explicit inequalities for polytopes associated with real flag varieties. Received: 8 February 1999 / in revised form: 25 October 1999 / Published online: 8 May 2000     

Malliavin calculus and decoupling inequalities in Banach spaces

We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided Lp-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.     

Regularity of solutions to a contact problem

We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.

     

A non-fixed point theorem for Hamiltonian lie group actions

We prove that, under certain conditions, if a compact connected Lie group acts effectively on a closed manifold, then there is no fixed point. Because two of the main conditions are satisfied by any Hamiltonian action on a closed symplectic manifold, the theorem applies nicely to such actions. The method of proof, however, is cohomological; and so the result applies more generally.