Duke Mathematical Journal

Topology of Riemannian submanifolds with prescribed boundary

Stephanie Alexander, Mohammad Ghomi, and Jeremy Wong

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Abstract

We prove that a smooth compact submanifold of codimension $2$ immersed in $\mathbf{R}^{n},n\geq 3,$ 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

Article information

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

Dates
First available in Project Euclid: 20 April 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1271783598

Digital Object Identifier
doi:10.1215/00127094-2010-018

Mathematical Reviews number (MathSciNet)
MR2654222

Zentralblatt MATH identifier
05708812

Subjects
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)

Citation

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. http://projecteuclid.org/euclid.dmj/1271783598.


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