The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

A geometric approach to error estimates for conservation laws posed on a spacetime

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov’s method, we derive an L¹ error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M),

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

Heat kernel expansions in vector bundles

Let M be a complete connected smooth (compact) Riemannian manifold of dimension n. Let Π:V→M be a smooth vector bundle over M. Let be a second order differential operator on M, where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M. Let k_t(−,−) be the heat kernel of V relative to L. In this paper we will derive an exact and an asymptotic expansion for k_t(x,y₀) where y₀ is the center of normal coordinates defined on M, x is a point in the normal neighborhood centered at y₀. The leading coefficients of the expansion are then computed at x=y₀ in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M, of the vector bundle V, and of the vector bundle section ? and its derivatives.We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean-Singer).     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrödinger-type differential expressions L(κ)=D²+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=^*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L²(E) and L²(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.     

Distortion of mappings and L q, p -cohomology

We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the L _q,p -cohomology of these manifolds. We apply these results to the L _q,p -cohomology of a manifold with a cusp.     

Morse homology for the heat flow – Linear theory

Consider the linear parabolic partial differential equation ${\mathcal {D}}_u\xi =0$ which arises by linearizing the heat flow on the loop space of a Riemannian manifold M. The solutions are vector fields along infinite cylinders u in M. For these solutions we establish regularity and a priori estimates. We show that for nondegenerate asymptotic boundary conditions the solutions decay exponentially in L² in forward and backward time. In this case ${\mathcal {D}}_u$ viewed as linear operator from the parabolic Sobolev space ${\mathcal {W}}^{1,p}$ to L^p is Fredholm whenever p > 1. We close with an L^p estimate for products of first order terms which is a crucial ingredient in the sequel 13 to prove regularity and the implicit function theorem. The results of the present text are the base to construct in 13 an algebraic chain complex whose homology represents the homology of the loop space.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λ_g: C^∞ (?M) → C^∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λ_gh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If g^t = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λ_g ^t = Λ_g + tΛ_? + o(t). We study the question: How do tensor fields ? look like for which Λ_? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.     

Riesz transform and perturbation

We prove that, on a complete noncompact Riemannian manifold with bounded geometry, the L^p boundedness of the Riesz transform, for p>2, is stable under a quasi-isometric and integrable change of metric. As an intermediate step, we treat the case of weighted divergence form operators in the Euclidean space.     

Asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary

In this article we discuss the asymptotic expansions of the zeta-determinants of Dirac Laplacians on a compact manifold with boundary when the boundary part is stretched. In [12] the author studied the same question under the assumption of no existence of L² - and extended L² -solutions of Dirac operators when the boundary part is stretched to infinite length. Therefore, the results in this article generalize those in [12]. Using the main results we obtain the formula describing the ratio of two zeta-determinants of Dirac Laplacians with the APS boundary conditions associated with two unitary involutions σ₁ and σ₂ on ker B satisfying Gσ_i = -σ_i G. We also prove the adiabatic decomposition formulas for the zeta-determinants of Dirac Laplacians on a closed manifold when the Dirichlet or the APS boundary condition is imposed on partitioned manifolds, which generalize the results in [10] and [11].     

The L p spectrum of Riemannian products

We prove that the L ^p spectrum of a Riemannian product M ₁×M ₂ coincides with the set theoretic sum of the L ^p spectra of M ₁ and M ₂ . Received: 13 June 2007     

Martingale transforms and L p -norm estimates of Riesz transforms on complete Riemannian manifolds

Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L ^p -inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L ^p -norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞. Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.     

Kähler-Ricci soliton typed equations on compact complex manifolds withC 1(M) > 0

As a generalization of Calabi’s conjecture for Kähler-Ricci forms, which was solved by Yau in 1977, we discuss the existence of Kähler-Ricci soliton typed equation on a compact Kähler manifold (M, g) with positive first Chem C₁ (M) > 0 as well as the uniqueness. For a given positively definite (1,1)-form Ω ∈ C₁ (M) of M and a holomorphic vector field X on M, we prove that there is a Kähler form ω in the Kähler class [ω_g] solving the Kähler-Ricci soliton typed equation if and only if, i) X is belonged to a reductive subalgebra of holomorphic vector fields and the imaginary part of X generates a compact one-parameter transformations subgroup of M; and ii) L_X Ω is a real-valued (1,1)-form. Moreover, the solution ω is unique in the class [ω_g].     

A note on the Hodge theory for functionals with linear growth

We study the Hodge decomposition of L ¹-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS]. Received: 19 November 1997 / Revised version: 18 May 1998