string theory , geometric Cauchy problem , timelike minimal surfaces
Abstract:
We investigate two-dimensional timelike surfaces ∑ in a spacetime (X,g). It is shown that orientable surfaces with two spacelike boundary components γ (homeomorphic to S¹) are necessarily of topological type [0,1] ∗ S¹. We treat the initial value problem of a string (known from physics) as a purely geometric problem: Find a minimal surface ∑ which is specified by an initial curve γ and by a distribution of timelike tangent planes along γ. We prove local existence and uniqueness of ∑ and also obtain global existence for special types (X, g). Global existence does not generally hold; we give a counter-example which can be interpreted as a string collapsing into a black hole.
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