Given a loop on a surface, its homotopy class can be specified as a word consisting of letters representing the homotopy group generators. One of the interesting problems is how to compute the shortest word for a given loop. This is an NP-hard problem in general. However, for a closed surface that allows a hyperbolic metric and is equipped with a canonical set of fundamental group generators, the shortest word problem can be reduced to finding the shortest loop that is homotopic to the given loop, which can be solved efficiently. In this paper, we propose an efficient algorithm to compute the shortest words for loops given on triangulated surface meshes. The design of this algorithm is inspired and guided by the work of Dehn and BirmanSeries. In support of the shortest word algorithm, we also propose efficient algorithms to compute shortest paths and shortest loops under hyperbolic metrics using a novel

Let <i>M</i> be a compact Khler manifold and <i>N</i> be a subvariety with codimension greater than or equal to 2. We show that there are no complete KhlerEinstein metrics on M - N . As an application, let <i>E</i> be an exceptional divisor of <i>M</i>. Then M - N cannot admit any complete KhlerEinstein metric if blow-down of <i>E</i> is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.

We consider the Einstein/Yang-Mills equations in 3+ 1 space time dimensions with SU (2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang/Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

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