MathSciDoc: An Archive for Mathematician ∫

We consider the coupled EinsteinDiracMaxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The stability and the properties of the ground state solutions are discussed for different values of the electromagnetic coupling constant. We find solutions even when the electromagnetic coupling is so strong that the total interaction is repulsive in the Newtonian limit. Our solutions are regular and well-behaved; this shows that the combined electromagnetic and gravitational self-interaction of the Dirac particles is finite.

We study three-dimensional Riemannian manifolds with nonnegative scalar curvature. We find new topological obstruction for such manifolds. Our method turns out to be useful in studying the positive mass conjecture in general relativity.

In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the rst case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the dening equations are transcendental. Indeed, even in the situation where the geometric object is homogeneous or algebraic, analytic methods often lead to important contributions. In this talk, we shall discuss a class of problems in differential geometry and the analytic methods that are involved in solving such problems.

IT IS well-known that if a compact group acts differentiably on a differentiable manifold, then this group must preserve some Riemannian metric on this manifold. From this point of view, we shall discuss certain facts about group actions on manifolds. In 01, we study the group of isometries of a non-compact manifold. We discover that if the group is non-compact, the manifold has to split as a product of the Euclidean space and another manifold. This gives some information on problem 33 of [5]. It also gives some information on the group of biholomorphic transformations of a complex manifold whose Bergman metric is not trivial. Then we find a topological obstruction for a non-compact manifold to admit an infinite group to act freely and properly discontinuously. Namely, we prove that the natural map from the de Rham cohomology group with compact support to the de Rham cohomology group without compact support is trivial when the first group has finite dimension. In $2, we obtain some topological obstructions for group actions by looking at the complex of invariant differential forms. We prove, for example, that if a compact group acts on a compact manifold with non-zero Euler number, then wI A*** A ut+,= 0 for all closed invariant l-forms wl,..., Ok+ 1 with k 2 the codimension of the principle orbit.(It may be interesting to note that the vanishing is on the form level so that any secondary obstruction should also vanish.) We also prove a topological version of this theorem for circle actions. An interesting corollary is that if a manifold is the connected sum of a torus and a compact manifold with Euler number# 2, then it does not admit any circle actions

As> iVinf^ b^ where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M with convex boundary. The techniques involve both the (discrete) heat kernels of graphs and improved estimates of the (continuous) heat kernels of Riemannian manifolds. We prove eigenvalue lower bounds for convex subgraphs of the form ce2/(dD (M)) 2 where e denotes the distance between two closest lattice points, D (M) denotes the diameter of the manifold M and c is a constant (independent of the dimension d and the number of vertices in 5, but depending on the how" dense" the lattice points are). This eigenvalue bound is useful for bounding the rates of convergence for various random walk problems. Since many enumeration problems can be approximated by considering random walks in convex subgraphs of some appropriate host graph, the eigenvalue inequalities here have many applications.

Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on certain complex Lie groups and their quotients. Both flat and non-flat cases are discussed in detail. This paper answers a question proposed by Andreas and Garcia-Fernandez in Comm Math Phys 332(3):13811383, 2014.

We report recent progress in the computation of conformal mappings from surfaces with arbitrary topologies to canonical domains. Two major computational methodologies are emphasized; one is holomorphic differentials based on Riemann surface theory and the other is surface Ricci flow from geometric analysis. The applications of surface conformal mapping in the field of engineering are briefly reviewed.

Many years ago, SY Cheng, P. Li and myself [1] gave an estimate of the heat kernel for the Laplacian. This was later improved by P. Li and myself in [2]. The key point of this later paper was a parabolic Harnack inequality that generalized an elliptic Harnack inequality that I developed more than twenty years ago.

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far.

We consider for <i>j</i>=, a spherically symmetric, static system of (2<i>j</i>+1) Dirac particles, each having total angular momentum <i>j</i>. The Dirac particles interact via a classical gravitational and electromagnetic field.