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Summary: TOPOLOGY OF NONNEGATIVELY CURVED HYPERSURFACES
WITH PRESCRIBED BOUNDARY IN Rn
STEPHANIE ALEXANDER, MOHAMMAD GHOMI, AND JEREMY WONG
Abstract. We prove that a smooth compact immersed submanifold of codimen-
sion 2 in Rn
, n 3, bounds at most finitely many topologically distinct compact
nonnegatively curved hypersurfaces. Analogous results for noncompact fillings
are obtained as well. On the other hand, we show that these topological finite-
ness theorems may not hold if the prescribed boundary is not sufficiently regular.
In particular we construct a simple closed differentiable and rectifiable curve in
R3
which bounds infinitely many topologically distinct smooth positively curved
surfaces. The proofs employ, among other methods, theorems of Gromov and
Perelman on Alexandrov spaces with curvature bounded below.
Contents
1. Introduction 2
2. The Example: Proof of Theorem 1.1 3
2.1. Overview 3
2.2. Construction of 4
2.3. Proof of Theorem 2.1 6
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