Duke Mathematical Journal

Topology of Riemannian submanifolds with prescribed boundary

Stephanie Alexander, Mohammad Ghomi, and Jeremy Wong

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We prove that a smooth compact submanifold of codimension 2 immersed in Rn,n3, bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman

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Duke Math. J., Volume 152, Number 3 (2010), 533-565.

First available in Project Euclid: 20 April 2010

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Zentralblatt MATH identifier

Primary: 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)


Alexander, Stephanie; Ghomi, Mohammad; Wong, Jeremy. Topology of Riemannian submanifolds with prescribed boundary. Duke Math. J. 152 (2010), no. 3, 533--565. doi:10.1215/00127094-2010-018. https://projecteuclid.org/euclid.dmj/1271783598

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