A geometric Cauchy problem for timelike minimal surfaces

Deck, Thomas

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URL: http://ub-madoc.bib.uni-mannheim.de/1722
URN: urn:nbn:de:bsz:180-madoc-17228
Document Type: Working paper
Year of publication: 1993
Publication language: German
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Classification: MSC: 83E30 53C42 35L70 ,
Subject headings (SWD): Zeichenkette , Cauchy-Anfangswertproblem , Minimalfläche , Lorentz-Mannigfaltigkeit
Keywords (English): 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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