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Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature,
Euclidean volume growth and quadratic curvature decay at infinity. By
comparison with capped spherical cones, we identify a precise borderline for
the Ricci curvature decay. Above this value, no complete area-minimizing
hypersurfaces exist. Below this value, in contrast, we construct examples.Comment: 31 pages. Comments are welcome
Minimal graphic functions on manifolds of non-negative Ricci curvature
We study minimal graphic functions on complete Riemannian manifolds \Si
with non-negative Ricci curvature, Euclidean volume growth and quadratic
curvature decay. We derive global bounds for the gradients for minimal graphic
functions of linear growth only on one side. Then we can obtain a Liouville
type theorem with such growth via splitting for tangent cones of \Si at
infinity. When, in contrast, we do not impose any growth restrictions for
minimal graphic functions, we also obtain a Liouville type theorem under a
certain non-radial Ricci curvature decay condition on \Si. In particular, the
borderline for the Ricci curvature decay is sharp by our example in the last
section.Comment: 38 page
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