Quadratic presentations and nilpotent Kähler groups

Research partially supported by National Science Foundation Grant DMS 9103966.     

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.     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

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.     

Besicovitch Covering Lemma,Hadamard manifolds,and zero entropy

It is proved that if the Besicovitch Covering Lemma is true on either a Hadamard manifold or a simply connected surface without focal points that covers a compact quotient, then the manifold is the Euclidean space. As a corollary, the vanishing of the topological entropy of a compact manifold of nonpositive curvature or of a compact surface without focal points is equivalent to the validity of the Besicovitch Covering Lemma on the universal covering space of the manifold. The author was partially supported by an NSF grant.     

The structure of area-minimizing double bubbles

We show that the least area required to enclose two volumes in ℝⁿ orS ⁿ forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝⁿ have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝⁿ. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝⁿ, based on an idea due to Brian White. Supported in part by NSF DMS-9409166.     

Geodesic transformations and harmonic spaces

We prove that a Riemannian manifold is harmonic if and only if there exists a divergence-preserving geodesic transformation with respect to each point which is not volume-preserving.     

Manifolds whose curvature operator has constant eigenvalues at the basepoint

Osserman conjectured that if the curvature operatorR of a Riemannian manifoldM has constant eigenvalues, thenM is locally a rank-1 symmetric space or is flat. The pointwise question is considerably more complicated. We present examples of Riemannian manifolds so thatR has constant eigenvalues at the basepoint, butR is not the curvature operator of a rank-1 symmetric space. Research partially supported by the NSF and IHES.     

Small excess and Ricci curvature

It is proved that a Riemanniann-manifold with Ricci curvature ≥ (n − 1) and a lower injectivity radius bound is a sphere provided the diameter is sufficiently close to π. The author was partially supported by the NSF and the Alfred P. Sloan Foundation.     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

On harmonic maps from stochastically complete manifolds

The main purpose of this article is to generalize a theorem about the size of minimal submanifolds in Euclidean spaces. In fact, we state and prove a non-existence theorem about harmonic maps from a stochastically complete manifold into a cone type domain. The proof is based on a generalized version of the maximum principle applied to the Lapalace-Beltrami operator on Riemannian manifolds. Received: 2 August 2007, Revised: 14 April 2008     

On a Step 2(k + 1) sub-Riemannian manifold

We compute the sub-Riemannian distance for a Step 2(k + 1) sub-Riemannian manifold from the origin to any given point. We characterize the number of sub-Riemannian geodesics between the origin and any other point.     

Growth of conjugacy classes of Schottky groups in higher rank symmetric spaces

Let X be a globally symmetric space of noncompact type and rank greater that one, and a Schottky group of axial isometries. Then is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homotopy with period less than t.     

Geometric analysis on quaternion ℍ-type groups

We construct some examples of ℍ-types Carnot groups related to quaternion numbers and study their geometric properties. We involve the Hamiltonian formalism to obtain the equations of geodesics and calculate the cardinality of geodesics joining two different points on these groups. We prove Kepler’s law and give a nice geometric interpretation of the length of geodesies.     

On the structure of pseudo-holomorphic discs with totally real boundary conditions

In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points, which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic trajectories for Lagrangian intersections in symplectic geometry. This paper is supported in part by NSF Grant DMS 9215011.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

A structure theorem of Dirac-harmonic maps between spheres

For an arbitrary Dirac-harmonic map (φ,ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N. On the basis, we could clarify all of non-trivial Dirac-harmonic maps from S ² to S ².     

Covering radii and paving diameters of Alexandrov spaces

Covering radii and paving diameters are defined, and the borderline case when cov_k X = π/2, k = 1,…,n + 1 and pav_k X = π/2, k = 1,…,n + 1 is studied (curv X ≥1, dim X = n).