Harmonic coordinates

Abstract: * Basic Properties of Harmonic Functions * Bounded Harmonic Functions * Positive Harmonic Functions * The Kelvin Transform * Harmonic Polynomials * Harmonic Hardy Spaces * Harmonic Functions on Half-Spaces * Harmonic Bergman Spaces * The Decomposition Theorem * Annular Regions * The Dirichlet Problem and Boundary Behavior * Volume, Surface Area, and Integration on Spheres * Harmonic Function Theory and Mathematica * References * Symbol Index * Index

Abstract: where dEλ is the projection valued measure associated with /^Δ\". A natural problem is to study the behavior of the explicit kernel kf(X)(xx, x2) representing /(/^Δ), in terms of the behavior of various geometric quantities on M. As a particularly important example we have the heat kernel E(xl9 x2, t) — ke-\\2t. By use of the local parametrix and the standard elliptic estimates, one can show that for / > 0, E(xλ9 JC2, 0 is a positive (symmetric) C function of JC1? x2, t which for fixed t and (say) %2> ι s * the domain of all positive powers of Δ as a function of xλ; see e.g. [9]. In works of Garding [19] and Donnelly [16], upper estimates for E(xu x2, t) (and its derivatives) were given under the assumption that M has bounded geometry. They showed that as x2 -> oo, the behavior of E{xλ, x2, t) is roughly similar to that of the e-p 2(xx,x2)/4 Euclidean heat kernel, — (p(xx, x2) denotes distance). Recall that (4ττ/) M is said to have bounded geometry if the injectivity radius i(x) of the

Abstract: We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass, scalar curvature and harmonic maps are described and the positive mass theorem for n-dimensional spin manifolds is proved.