Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above
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In this paper, we show a local energy convexity of W1,2 maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi’s width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.
KeywordsWidth Sweepout Min-max Closed geodesic Alexandrov space of curvature bounded from above
Mathematics Subject Classification (2000)53C23 58E10 53C22
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