Showing papers in "Journal of Geometric Analysis in 2003"
TL;DR: In this paper, the authors studied codimension 1 rectifiable sets in Carnot groups and extended classical De Giorgi's rectifiability and divergence theorems to the setting of step 2 groups.
Abstract: In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability and divergence theorems to the setting of step 2 groups Related problems in higher step Carnot groups are discussed, pointing on new phenomena related to the blow up procedure
210 citations
TL;DR: In this paper, the authors improved a result of L.A. Caffarelli and N.M. Riviere by showing that every connected component of the interior of the coincidence set has at most two singular points.
Abstract: We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Riviere: 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 uλ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points.
69 citations
TL;DR: In this article, the scalar curvature and the boundary mean curvature of the standard half-three sphere were computed using blow-up analysis and minimax arguments, and some existence and compactness results were proved.
Abstract: We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half-three sphere, by deforming conformally its standard metric. Using blow-up analysis techniques and minimax arguments, we prove some existence and compactness results.
63 citations
TL;DR: In this article, the authors define lower and upper weighted densities of a weighted wavelet system with respect to the geometry of the affine group, and prove necessary conditions on the system to possess frame bounds.
Abstract: If ψ ∈ L2(R), Λ is a discrete subset of the affine groupA =R+ ×R, and w: Λ →R+ is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is\(\mathcal{W}(\psi ,\Lambda ,\omega ) = \{ \omega (a,b)^{1/2} a^{ - 1/2} \psi (\frac{x}{a} - b):(a,b) \in \Lambda \} \). In this article we define lower and upper weighted densities Dw−(Λ) and Dw+(Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound, then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses a lower frame bound and Dw+(Λ−1) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems.
55 citations
TL;DR: In this article, a Gabor orthonormal basis for L2(ℝ) was constructed, such that the generating function g ∈L2(∄") satisfies the condition: ∫℘|g(x)|2(1+|x|consuming 2(2 + \left| \xi \right|^B )/log ^d (2 + ǫd (ǫ)d\xi < ∞ ).
Abstract: For any positive real numbers A, B, and d satisfying the conditions
$$\frac{1}{A} + \frac{1}{B} = 1$$
, d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫ℝ|g(x)|2(1+|x|
A
)/log
d
(2+|x|)dx < ∞ and
$$\int_{\hat {\mathbb{R}}} {\left| {\hat g(\xi )} \right|^2 (1 + \left| \xi \right|^B )/\log ^d (2 + \left| \xi \right|)d\xi< \infty } $$
.
51 citations
TL;DR: In this paper, it was shown that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp.
Abstract: The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain group-invariant polynomials (related to Lucas polynomials) achieve the bound. To establish the bound the authors introduce a graph-theoretic approach that requires determining the number of sinks in a directed graph associated with the quotient polynomial. The proof also relies on a result of the first author that expresses all proper polynomial holomorphic mappings between balls in terms of tensor products.
51 citations
TL;DR: The main result of as mentioned in this paper is the existence and uniqueness of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternion strictly pseudoconvex bounded domains.
Abstract: The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in ℍ
n
. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].
42 citations
TL;DR: In this article, the authors compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds, and give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3 and a length of a closed geodesic that occurs in one manifold but cannot occur in the other.
Abstract: We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials.
34 citations
TL;DR: In this article, the Newlander-Nirenberg theorem is extended to manifolds with almost complex structures that have somewhat less Lipschitz regularity than the integrable case.
Abstract: We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.
33 citations
TL;DR: In this article, the Euclidean Sobolev inequality for Riemannian manifolds with dimension n ≥ 5 has been shown to be attainable for any compact manifold.
Abstract: 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),
$$\left\| u \right\|_{2^\sharp }^2 \leqslant A\left\| {\Delta _g u} \right\|_2^2 + B\left\| u \right\|_{H_1^2 }^2 $$
where 2#=2n/(n−4) is critical, and\(\left\| \cdot \right\|_{H_1^2 } \) is the usual norm on the Sobolev space H 1 2 (M) consisting of functions in L2(M) whose derivatives of order one are also in L2(M). The sharp constant A in this inequality is K 0 2 where K0, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality\(\left\| u \right\|_{2^\sharp } \leqslant K\left\| {
abla u} \right\|_2 \). We prove in this article that for any compact Riemannian manifold, A=K 0 2 is attained in the above inequality.
26 citations
TL;DR: In this paper, it was shown that when the self-intersection number of the curve is sufficiently negative, the fibration is equivalent to the linear one defined in the normal bundle to the curve.
Abstract: The aim of this article is to study fibered neighborhoods of compact holomorphic curves embedded in surfaces. It is shown that when the self-intersection number of the curve is sufficiently negative the fibration is equivalent to the linear one defined in the normal bundle to the curve. The obstructions to equivalence in the general case are described.
TL;DR: In this paper, the conformal class of a Hermitian metric g on a compact almost complex manifold (M2m, J) consists entirely of metrics that are Hermitians with respect to J, and it is shown that (2m−1)sJ−s=2(2m −1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g.
Abstract: The conformal class of a Hermitian metric g on a compact almost complex manifold (M2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature sJ. We ask if the conformal class of g carries a metric with constant sJ, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)sJ−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere\(\mathbb{S}^6 \) equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.
TL;DR: In this paper, it was shown that π can be approximated by an isotopic immersion π1: S→X whose image has a basis of open Stein neighborhood in X which are homotopy equivalents to π(S).
Abstract: Let S be a closed connected real surface and π: S→X a smooth embedding or immersion of S into a complex surface X. We denote by I(π) (resp. by I±(π) if S is oriented) the number of complex points of π (S)∪X counted with algebraic multiplicities. Assuming that I(π)≤0 (resp. I±(π)≤0 if S is oriented) we prove that π can be C0 approximated by an isotopic immersion π1: S→X whose image has a basis of open Stein neighborhood in X which are homotopy equivalents to π1 (S). We obtain precise results for surfaces in\(\mathbb{C}\mathbb{P}^2 \) and find an immersed symplectic sphere in\(\mathbb{C}\mathbb{P}^2 \) with a Stein neighborhood.
TL;DR: In this article, the existence d'estimations optimales for the metrique de Kobayashi dans les domaines pseudoconvex de type fini deC 2 a un principe de Bloch asymptotique is established.
Abstract: Nous ramenons l'existence d'estimations optimales pour la metrique de Kobayashi dans les domaines pseudoconvexes de type fini deC
2 a un principe de Bloch asymptotique. Nous etablissons ce principe en combinant la methode de renormalisation utilisee par Gromov dans le contexte des applications harmoniques aux techniques de dilatation des coordonnees. Cecifournit une preuve totalement elementaire d'un resultat de Catlin particulierement utile dans l'etude des questions de prolongement et de rigidite d'applications holomorphes.
TL;DR: For a compact manifold M of dimension n ≥ 4, the Yamabe constant YC(M) and the Ln/2-norm WC(m) of the Weyl curvature were studied in this paper, where the image of the mapYW:C→(YC(M), WC(M))∈R 2 near the line {(Y(M, w)|w ∈R}.
Abstract: For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant YC(M) and the Ln/2-norm WC(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant YC(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant WC(M) is arbitrarily large. We study the image of the mapYW:C→(YC(M), WC(M))∈R
2 near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kahler surfaces of Kodaira dimension 0, 1 or 2.
TL;DR: A survey of the development of methods in the theory of partial differential equations for the study of the Levi and Cousin problems in complex analysis is given in this paper, with the background in Hodge theory and with early unsuccessful attempts to exploit the Bergman kernel.
Abstract: The purpose of this paper is to give a historical survey of the development of methods in the theory of partial differential equations for the study of the Levi and Cousin problems in complex analysis. Success was achieved by the mid 1960's but we begin further back, with the background in Hodge theory and with early unsuccessful attempts to exploit the Bergman kernel. Some examples of later date illustrating the usefulness of such methods are also given.
TL;DR: In this paper, Dahlberg's theorems on the mutual absolute continuity of harmonic and surface measures are extended to compact polyhedral domains of ℝn, and the Dirichlet problem for Laplace's equation with data taken in Lp spaces p > 2 − δ is solved.
Abstract: B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in Lp spaces p > 2 − δ are extended to compact polyhedral domains of ℝn. Consequently, for q < 2 + δ, Dahlberg’s reverse Holder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.
TL;DR: In this paper, the same authors construct examples of pairs of surfaces that share the same poles (with multiplicity) of the scattering operatorm, and that are particularly simple in one or more senses.
Abstract: We construct examples of pairs of surfaces that share the same poles (with multiplicity) of the scattering operatorm, and that are particularly simple in one or more senses. In particular, we find pairs of isoscattering surfaces of small genus with a small number of ends. We also give examples of congruence surfaces which are isoscattering.
TL;DR: In this article, a lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric was obtained.
Abstract: In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum.
TL;DR: In this paper, the authors established existence of a solution to the 2-phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hn−1 Hausdorff measure and a normal in a measuretheoretic sense almost everywhere.
Abstract: In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hn−1 Hausdorff measure and a normal in a measuretheoretic sense Hn−1 almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact that the free boundary is locally a C1,α surface near Hn−1-a.e. point.
TL;DR: In this paper, it was shown that ∫exp(iR(t)) dt/t has a uniform bound with a bound depending only on the degrees of P and Q and not on their coefficients.
Abstract: Let R(t)=P(t)/Q(t) be a quotient of real polynomials. We show that ∫exp(iR(t)) dt/t has a uniform bound with a bound depending only on the degrees of P and Q and not on their coefficients. Also LP estimates are obtained for certain associated singular integral operators.
TL;DR: In this paper, the authors introduced a new method to construct large classes of wavelets of the Hardy space H 2 (ℝ) and symmetric wavelet sets of L 2 (∆) and H 2 ∆ (ↆ) with Fourier transform even and not vanishing in any neighborhood of origin.
Abstract: 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.
TL;DR: In this article, the 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.
Abstract: 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.
TL;DR: In this paper, a pair of conformally equivalent isospectral Riemannian metrics ϕ1g and ϕ2g 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 was constructed.
Abstract: We construct pairs of conformally equivalent isospectral Riemannian metrics ϕ1g and ϕ2g 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 ϕ1 and ϕ2 are isospectral potentials for the Schrodinger operator ħ2\gD + \gf. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.
TL;DR: In this paper, it was shown that a proper holomorphic mapping between bounded domains D, D′ in ℂ2 extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings.
Abstract: Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ2.Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ2.
TL;DR: In this article, it was shown that a function h is a functional Gabor frame multiplier if and only if it is unimodular and h(t)/h(t + \frac{1}{b} is a-periodic.
Abstract: A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and\(h(t)/h(t + \frac{1}{b})\) is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L∞(ℝ) such that {wgmn} (resp. ωψk,ℝ) is a normalized tight frame.
TL;DR: In this article, the authors studied a two-phase free boundary problem in which the speed of the free boundary depends also on its curvature and proved that the solution and free boundary are classical.
Abstract: We study a two-phase free boundary problem in which the speed of the free boundary depends also on its curvature. It is assumed that the free boundary is Lipschitz and it is proved that the solution as well as the free boundary are classical.
TL;DR: Theorems of Wiman and Arima about entire holomorphic functions of a complex variable are generalized to the case of n-subharmonic functions on spherical n-dimensional cones.
Abstract: Theorems of Wiman and Arima about entire holomorphic functions of a complex variable are generalized to the case of n—subharmonic functions on spherical n-dimensional cones.
TL;DR: In this paper, a singularly perturbed semi-linear elliptic PDE with a bistable potential on a closed Riemannian surface was studied and it was shown that the transition region converges in the sense of varifolds to an embedded (multiple) curve with constant geodesic curvature.
Abstract: We study a singularly perturbed semi-linear elliptic PDE with a bistable potential on a closed Riemannian surface. We show that the transition region converges in the sense of varifolds to an embedded (multiple) curve with constant geodesic curvature.
TL;DR: In this paper, the authors studied Riemannian foliations with complex leaves on Kahler manifolds, where the tensor T, the obstruction to the foliation, is interpreted as a holomorphic section of a certain vector bundle.
Abstract: We study Riemannian foliations with complex leaves on Kahler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.