An index formula on manifolds with fibered cusp ends

We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra that models fibered cusps at infinity. Using tracelike functionals that generate the 0-dimensional Hochschild cohomology groups we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior of X and a term that comes from the boundary. This leads to an abstract answer to the index problem formulated in [11]. We give a more precise answer for firstorder differential operators when the base of the boundary fiber bundle is S¹. In particular, for Dirac operators associated to a metric of the form near ∂X = {x = 0} with twisting bundle T we obtain

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

Isoscattering deformations for complete manifolds of negative curvature

We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.     

Harmonic functions under quasi-isometry

Given a complete Riemannian manifold (M, g) with nonnegative sectional curvature outside a compact subset. Let h be another Riemannian metric which is uniformly equivalent to g. It was shown that the dimension of the space of bounded harmonic functions on (M, h) is finite and is the same as of that under metric g, and the dimension of the space spanned by nonnegative harmonic functions on (M, h) is also finite and is the same as of that under metric g. Moreover, bases were constructed for both spaces on (M, h) and precise estimates were established on the asymptotic behavior at infinity for those basic functions.     

Small eigenvalues of geometrically finite manifolds

Let M be a complete geometrically finite manifold of bounded negative curvature, infinite volume, and dimension at least 3.We give both a lower bound for the bottom of the spectrum of M and an upper bound for the number of the small eigenvalues of M. These bounds only depend on the dimension, curvature bounds and the volume of the oneneighborhood of the convex core.     

On estimates of biharmonic functions on Lipschitz and convex domains

Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in ℝⁿ. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the L^p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L^p Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4.     

Isospectral potentials and conformally equivalent isospectral metrics on spheres,balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics ?1g and ?2g on spheres Sⁿ and balls Bⁿ⁺¹ for certain dimensions n, the smallest of which is n=7, and on certain compact simple Lie groups. In the case of Lie groups, the metric g is left-invariant. In the case of spheres and balls, the metric g not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds (M, g) we also show that the functions ?₁ and ?₂ are isospectral potentials for the Schrödinger operator ?₂\gD + \gf. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.     

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].     

Extrapolation of Weighted norm inequalities for multivariable operators and applications

Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces. Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose indices lie on a line. Applications are given in the context of multilinear Calderón-Zygmund operators. Vector-valued inequalities are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the Marcinkiewicz and Zygmund theorem on ℓ²-valued extensions of bounded linear operators is also obtained.     

Entropy and reduced distance for Ricci expanders

Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂_gij/∂t = − 2R_ij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W₊ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ₊ and v₊. The forward reduced volume θ₊ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/t^n/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like t^n/2(maximal volume growth) then W⁺, θ⁺ and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.     

The entropy formula for linear heat equation

We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.     

The local part and the strong type for operators related to the Gaussian measure

In this work we present several theorems which imply the weak type 1 with respect to the Gaussian measure for the so-called local part of certain operators associated with the Ornstein-Uhlenbeck semigroup. Particular cases of these operators are the Riesz transforms of any order and the Littlewood-Paley square function. Also, we study general results based on the “size” of the operator which ensure the strong type 1 <p < ∞of both the local and global parts.     

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds

Existence of solution for semilinear problem with the Laplace-Beltrami operator on non-compact Riemannian manifolds with rich symmetries is proved by concentration compactness based on actions of the manifold's isometry group.     

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),

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

The heat flow and harmonic maps between complete manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application, we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing energy density.     

The fundamental solution on manifolds with time-dependent metrics

In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u, on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold. We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel.     

On the large time behavior of the heat kernels of quasiperiodic differential operators

We prove an analog of the Berry-Esseen estimate for the heat kernel of second order elliptic differential operators with quasiperiodic coefficients. As an application of this result, we prove the L^p boundedness of the associated Riesz transform operators.     

Large classes of minimally supported frequency wavelets of<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ) and<Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>(ℝ)

We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H ² (ℝ)and symmetric MSF wavelets of L ² (ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets of L ² (ℝ)and H ² (ℝ).We also enumerate some of the symmetric wavelet sets of L ² (ℝ)and all wavelet sets of H ² (ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L ₂ (ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin.