A differentiable sphere theorem on manifolds with reverse volume pinching

Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

On eigenvalue pinching in positive Ricci curvature

We shall show that for manifolds with Ric≥n−1 the radius is close to π iff the (n+1)st eigenvalue is close to n. This extends results of Cheng and Croke which show that the diameter is close to π iff the first eigenvalue is close to n. We shall also give a new proof of an important theorem of Colding to the effect that if the radius is close to π, then the volume is close to that of the sphere and the manifold is Gromov-Hausdorff close to the sphere. From work of Cheeger and Colding these conditions imply that the manifold is diffeomorphic to a sphere. Oblatum 29-V-1998 & 4-II-1999 / Published online: 21 May 1999     

L p -pinching and the geometry of compact Riemannian manifolds

We prove a Harnack-type inequality inf|S|/sup|S|>1?ε(W, M, V) satisfied by the sections of a Riemannian vector bundleW lying in the kernel of a Schrödinger operator ∨*∨+V underL ^p -pinching assumptions on the potentialV and derive various topological and geometric consequences. For instance, we prove a fibration theorem which gives a classification of almost non-negatively curved compact manifolds by the first Betti number. In the case of almost non-positively curved compact manifolds, we prove that the minimal volume must vanish whenever the isometry group is not finite and give conditions implying that it is abelian.     

A proof of the Faber-Krahn inequality for the first eigenvalue of thep-Laplacian

Summary In this work we present an elementary proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian on bounded domains in ℝⁿ. Let λ₁ be the first eigenvalue and λ ₁ ^* be the first eigenvalue for the ball of the same volume. Then we show that λ₁=λ ₁ ^* iff the domain is a ball. Our proof makes considerable use of the corresponding Talenti's inequality and some well known properties of the first eigenfunction. Entrata in Redazione il 20 aprile 1998.     

Large continuum,oracles

Our main theorem is about iterated forcing for making the continuum larger than ℵ₂. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ₁,ℵ₂ by λ, λ ⁺ (starting with λ = λ ^<λ > ℵ₁). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ ⁺ but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.     

Complete settling of the multiplier conjecture for the case of n=3pr

In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n = 3n₁. We prove that in the case of n = 3n₁ Second multiplier theorem remains true if the assumption “n₁ > λ” is replaced by “(n₁, λ) = 1”. Consequentially we prove that if we let D be a (v, k, λ)-difference set in an abelian group G, and n = 3p^r for some prime p, (p,v) = 1, then p is a numerical multiplier of D.     

On the Norm of the Fourier—Gegenbauer Projection in Weighted L p Spaces

We extend the results of Pollard [4] and give asymptotic estimates for the norm of the Fourier—Gegenbauer projection operator in the appropriate weighted L _p space. In particular, we settle the question of whether the projection is bounded for p=(2λ+1)/λ and p=(2λ+1)/(λ+1) , where λ is the index for the family of Gegenbauer polynomials under consideration. March 19, 1997. Date revised: June 3, 1998. Date accepted: August 1, 1998.     

Abel-Lidskii bases in the non-self-adjoint inverse boundary problem

Let M be a manifold with boundary δM≠Ф. Let A be a second-order elliptic partial differential operator given on M. Denote by R_λ(x, y), x, y∈M, λε \σ(A) the Schwartz kernel of (A−λI)⁻¹. Consider the Gel'fand inverse boundary problem of the reconstruction of (M, A) via a given R_λ(x, y), x, y∈δM,λε . We prove that if the principal symbol of A satisfies some geometric condition (the Bardos-Lebeau-Rauch condition), then these data determine M uniquely, and they determine A to within the group of generalized gauge transformation on M. The above-mentioned geometric condition means, roughly speaking, that any geodesic (in the metric generated by A) leaves M. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 161–190. Translated by Ya. Kurylev.     

Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space

In this article, we prove new pinching theorems for the first eigenvalue λ₁(M) of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for λ₁(M) in terms of higher order mean curvatures H _k . We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost-isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost-Einstein is diffeomorpic and almost-isometric to a standard sphere.      

On a class of semilinear weakly hyperbolic equations

Abstract  In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

On the number of singularities for the obstacle problem in two dimensions

We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière: we state that every connected component of the interior of the coincidence set has at most N ₀ singular points, where N ₀ is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free boundary of the solution u_λ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points. Dedicated to Henri Berestycki and Alexis Bonnet.     

On the multiplier conjecture

The Multiplier Theorem is a celebrated theorem in the Design theory. The conditionp>λ is crucial to all known proofs of the multiplier theorem. However in all known examples of difference sets μ _p . is a multiplier for every primep with (p, v)=1 andp‖n. Thus there is the multiplier conjecture: “The multiplier theorem holds without the assumption thatp>λ”. The general form of the multiplier theorem may be viewed as an attempt to partially resolve the multiplier conjecture, where the assumption “p>λ” is replaced by “n ₁>λ”. Since then Newman (1963), Turyn (1964), and McFarland (1970) attempted to partially resolve the multiplier conjecture (see [7], [8], [9]). This paper will prove the following result using the representation theory of finite groups and the algebraic number theory: LetG be an abelian group of orderv,v ₀ be the exponent ofG, andD be a (v, k, λ)-difference set inG. Ifn=2_{n 1}, then the general form of the multiplier theorem holds without the assumption thatn ₁>λ in any of the following cases:

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions. In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact Riemannian manifolds.     

Cubic Diophantine Inequalities

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ _i ≥ 1. In this paper, we prove there are integers x ₁,..., x ₇ such that the inequalities |λ₁ x ³ ₁ + λ₂ x ³ ₂ + ⋯ + λ₇ x ³ ₇| < 1 and hold simultaneously. Received November 18, 1997, Accepted October 23, 1998     

Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities

In this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on a real parameter λ, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case λ < λ ₁ and λ ≥ λ ₁, where λ₁ is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality.     

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

We investigate the recursive sequence Z _n : =  max {Z _n − 1,λ(Z _n − 1)X _n } where X _n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z _n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z _n . The resulting limit is a Gumbel distribution.     

“Gap 1” two-cardinal principles and the omitting types theorem for ℒ(Q)

Forλ a strong limit singular cardinal, and more generally forλ > 2^cofλ , we prove the equivalence of a number of model theoretic and combinatorial conditions, including the ℒ(Q)-completeness theorem for theλ ⁺-interpretation, an omitting types theorem for ℒ(Q) in theλ ⁺-interpretation, and a weak form of Jensen’s principle □_λ. Research supported in part by the United States-Israel Binational Science Foundation. The author thanks the referee for a thorough revision of the paper. Publication #269.     

Contributions to the Hypergeometric Function Theory of Heckman and Opdam: Sharp Estimates, Schwartz Space, Heat Kernel

Under the assumption of positive multiplicity, we obtain basic estimates of the hypergeometric functions F _λ and G _λ of Heckman and Opdam, and sharp estimates of the particular functions F ₀ and G ₀. Next we prove the Paley–Wiener theorem for the Schwartz class, solve the heat equation and estimate the heat kernel. Received: June 2006 Revision: December 2006 Accepted: March 2007     

Minimal entropy and simplicial volume

We prove that MinEnt (Y) ∥Y∥ = MinEnt(X) ∥X∥, for manifolds Y whose fundamental group is a subexponential extension of the fundamental group of some negatively curved, locally symmetric manifold X. This is a particular case of a more general result holding for an arbitrary representation ρ : π₁ (Y) →π₁ (X), which relates the minimal entropy and the simplicial volume of X to some invariants of the couple (Y, ker (ρ)). Then, we discuss some applications to the minimal volume problem and to Einstein metrics. Received: 23 December 1998