Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Maximal estimates on measure spaces for weak-type multipliers

We describe sufficient conditions for transferring from locally compact abelian groups to measure spaces the weak-type bounds of maximal operators defined by multipliers of weak type. This leads to homomorphism theorems for maximal multiplier operators. Communicated by Guido Weiss     

Gradient Young measures generated by sequences in Sobolev spaces

Oscillatory properties of a weak convergent sequence of functions bounded inL ^p , 1 ≤p ≤ ∞, may be summarized by the parametrized measure it generates. When such a measure is generated by the gradients of a sequence of functions bounded inH ^1,p , it must have special properties. The purpose of this paper is to characterize such parametrized measures as the ones that obey Jensen’s inequality for all quasiconvex functions with the appropriate growth at infinity. We have found subtle differences between the casesp < ∞ andp = ∞. A consequence is that any measure determined by biting convergence is in fact generated by a sequence convergent in a stronger sense. We also give a few applications. Research groupTransitions and Defects in Ordered Materials, funded by the NSF, the AFOSR, and the ARO. The work of the second author is also supported by DGICYT (Spain) through “Programa de Perfeccionamiento y Movilidad del Personal Investigador” and through Grant PB90-0245.     

An explicit computation of the Bergman kernel function

We give an explicit computation of the Bergman kernel function on the domain

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.     

Local analyticity for the [`(?)]\overline \partial -Neumann problem in completely decoupled pseudoconvex domains-Neumann problem in completely decoupled pseudoconvex domains

In this paper we prove local analyticity of solutions to the -Neumann problem up to the boundary of rigid, completely decoupled pseudoconvex domains with real-analytic boundary. These are domains that are locally of the form Imw > Σ |h _k (z _k )|² with eachh _k holomorphic and vanishing only at 0. As in those earlier papers, we use purelyL ² methods and must construct a special holomorphic vector fieldM and then use carefully balanced polynomials inM to localize high powers ofT = ∂/∂t effectively, wheret = Rew.     

Unique continuation theorems for the\bar \partial - Operator and applications

We formulate a unique continuation principle for the inhomogeneous Cauchy-Riemann equations near a boundary pointz ₀ of a smooth domain in complex euclidean space. The principle implies that the Bergman projection of a function supported away fromz ₀ cannot vanish to infinite order atz ₀ unless it vanishes identically. We prove that the principle holds in planar domains and in domains where the problem is known to be analytic hypoelliptic. We also demonstrate the relevance of such questions to mapping problems in several complex variables. The last section of the paper deals with unique continuation properties of the Szegő projection and kernel in planar domains. Research supported by NSF Grant DMS-8922810.     

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S² V S² V ... →N. The main result is that the images of these renormalized maps converge in L^1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.     

Generalized reinhardt domains

In this paper we investigate a class of Lie group actions on , the so-calledpolar actions, that naturally generalize the standard actions. For a domain invariant under such an action (i.e., a generalized Reinhardt domain) we characterize the invariant plurisubharmonic functions and determine the envelope of holomorphy in geometric terms. For a generalized Reinhardt domain containing the origin of we also compute its automorphism group. Supported in part by NSF Grant 8602020