Boundary Behavior of Harmonic Functions on Manifolds

In this paper, we mainly set up a kind of representation theorem of harmonic functions on manifolds with Ricci curvature bounded below and study non-tangential limits of harmonic functions.     

Locally Minimizing Smooth Harmonic Maps from Asymptotically at Manifolds into Spheres

By minimizing the so-called relative energy, we show that there exists a family of locally minimizing smooth harmonic maps from asymptotically flat manifolds into the standard sphere.     

调和函数和完备流形的结构

主要通过讨论调和函数来研究完备流形的几何性质,并推广了[1,9]中的结果．     

一类完备Riemann流形上的有界调和函数

本文我们将对一类完备Ｒｉｅｍａｎｎ流形上的有界调和函数所组成的线性空间的维数的上界进行估计，同时给出了一个关于测地球体积的Ｂｉｓｈｏｐ－Ｇｒｏｍｏｖ型体积比较定理。     

一类完备Riemann流形的正调和函数

本文主要讨论一类完备Ｒｉｅｍａｎｎ流形上的调和函数所组成的线性空间．推广了Ｐ．Ｌｉ及Ｌ．Ｆ．Ｔａｍ^{[5], [7]}和和Ｇｒｅｅｎｅ－Ｗｕ^[3]中的结果．     

Existence of Harmonic Functions with Finite Energy on Complete Riemannian Manifolds

Let M be a noncompact complete Riemannian manifold. We consider the existence of harmonic functions with |∇u| ∈ L^p(M).     

Radiation Fields on Asymptotically Euclidean Manifolds

Abstract

F.G. Friedlander introduced the notion of radiation fields for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation fields.     

The Einstein-Scalar Field Constraints on Asymptotically Euclidean Manifolds

By using the conformal method, solutions of the Einstein-scalar field gravitational constraint equations are obtained. Handling scalar fields is a bit more challenging than handling matter fields such as fluids, Maxwell fields or Yang-Mills fields, because the scalar field introduces three extra terms into the Lichnerowicz equation, rather than just one. The proofs are constructive and allow for arbitrary dimension (> 2) as well as low regularity initial data.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

Harmonic Manifolds and Tubes

Csikós and Horváth (J Lond Math Soc (2) 94(1):141–160, 2016) showed that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed only for tubes about geodesic segments. As a consequence, we obtain similar characterizations of harmonic manifolds in terms of the total mean curvature and the total scalar curvature of tubular hypersurfaces about curves. We find simple formulae expressing the volume, total mean curvature, and total scalar curvature of tubular hypersurfaces about a curve in a harmonic manifold as a function of the volume density function.     

Computing Asymptotic Invariants with the Ricci Tensor on Asymptotically Flat and Asymptotically Hyperbolic Manifolds

We prove in a simple and coordinate-free way the equivalence between the classical definitions of the mass or of the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.     

Resolvent Estimates for the Laplacian on Asymptotically Hyperbolic Manifolds

Combining results of Cardoso-Vodev [6] and Froese-Hislop [9], we use Mourre’s theory to prove high energy estimates for the boundary values of the weighted resolvent of the Laplacian on an asymptotically hyperbolic manifold. We derive estimates involving a class of pseudo-differential weights which are more natural in the asymptotically hyperbolic geometry than the weights used in [6]. submitted 28/04/05, accepted 26/09/05     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

Sectional Curvature Rigidity of Asymptotically Locally Hyperbolic Manifolds

This paper presents a rigidity result of real hyperbolic quotients inanalogy to Min-Oo's result in Math. Ann. 285(4) (1989),527–539, but without the spin condition. In order to prove this,we use special Killing forms on the exterior form bundle. Moreover, wemake an assumption on the sectional curvature to obtain the necessaryeigenvalue estimates of the curvature endomorphism in theBochner–Weitzenböck formula of ^k M.     

Integrable Harmonic Functions on

A class of radial measuresμon ⁿis defined so that integrable harmonic functionsfon ⁿmay be characterized as solutions of convolution equationsf*μ=f. In particular we show thatf*e^{−2*π |x|}) is harmonic if and only ifn<9.     

Invariant Metrics on Cohomogeneity One Manifolds

Let G be one of the connected subgroups of the orthogonal group of ⁿ which acts transitively on the unit sphere S ^n–1. We get the necessary and sufficient condition for G-invariant metrics g on ⁿ \{0} to be extendend to the origin. For n=2 this is a classical result of Berard–Bergery. The curvature tensor and the sectional curvature of any such Riemannian G-manifold ( ⁿ , g) are described in terms of the length of the Killing vector fields, as well as the second fundamental form of the regular orbits G(P)=S ^n–1. As an application we describe all G-invariant metrics which are Kähler, hyperKähler or have constant principal curvatures. Some of these results are generalized to the case of any cohomogeneity one G-manifold which, in a neighbourhood of a singular orbit, can be identified with a twisted product.     

Non-existence for Harmonic Maps on Complete Noncompact Manifolds

In this paper, we prove a nonexistence theorem on harmonic maps. This generalizes the well-known Liouville-type theorem on harmonic maps due to S.Y. Cheng and H.I Choi.     

Harmonic Maps from Complete Noncompact Manifolds

ln this paper we prove some general existence theorems of harmonic maps from complete noncompact manifolds with tho positive lower bounds of spectrum into convex balls. We solve the Dirichlet problem in classical domains and some special complete noncompact manifolds for harmonic maps into convex balls. We also study the existence of harmonic maps from some special complete noncompact manifolds into complete manifolds with nonpositive sectional curvature which are not simply connected.     

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action. Submitted: March 16, 2007. Accepted: June 14, 2007.