Topology of Riemannian submanifolds with prescribed boundary
In this interesting paper, the authors prove that a smooth compact submanifold of codimension \(2\) immersed in \(\mathbb R ^{n}, \, n \geq 3,\) bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This solves a question of \textit{B. Guan} and \textit{J. Spruck} [Commun. Pure Appl. Math. 57, No. 10, 1311--1331 (2004; Zbl 1066.53109)] related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are also obtained. They also show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular. The methods of the proofs include 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.
scientific article; zbMATH DE number 5708812
scientific article