Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature

Abstract

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.1991 Mathematics Subject Classification. 53A10; 53C21; 58E12. Key words and phrases. Minimal hypersurface, non-negative Ricci curvature, tangent cone at infinity.

Minimal Graphic Functions on Manifolds of Nonnegative Ricci Curvature

“…Analogously to [21], we prove that for any scaling sequence of a minimal graph M in † R there exists a subsequence that converges to an area-minimizing cone T in † 1 R, where † 1 is some tangent cone at infinity (not necessarily unique) of † satisfying conditions (C1), (C2), and (C3). Analogously to [21], we prove that for any scaling sequence of a minimal graph M in † R there exists a subsequence that converges to an area-minimizing cone T in † 1 R, where † 1 is some tangent cone at infinity (not necessarily unique) of † satisfying conditions (C1), (C2), and (C3).…”

“…By [21] and the maximum principle, we have (6.5) ju j j c j jx nC1 jr p 1 in B j : If w j is a solution of (6.4) with boundary d j x nC1 r p 1 and 0 < d j < c j , we have ju j j > jw j j on B j \ fx nC1 ¤ 0g. By [21] and the maximum principle, we have (6.5) ju j j c j jx nC1 jr p 1 in B j : If w j is a solution of (6.4) with boundary d j x nC1 r p 1 and 0 < d j < c j , we have ju j j > jw j j on B j \ fx nC1 ¤ 0g.…”

“…The above three conditions have been used in [21] to study minimal hypersurfaces in such manifolds. n r n for all x 2 † and r > 0; where !…”

“…However, our proof here is more complicated than [21] as the ambient manifold † R no longer satisfies condition (C3) unless † is flat. This also suffices to estimate the measure of a "bad" set and employ stability arguments, as in [21], to eliminate the unbounded situation of jDuj; when the lower bound Ä for the curvature decay (see below) is large, that is, when the curvature can only slowly decay to 0 at infinity (more precisely, the curvature decay is quadratic, but the value of the lower bound must be above some critical threshold; actually the fact that that threshold is sharp was shown in the last section). However, since we lack Sobolev and Neumann-PoincarK e inequalities on minimal graphs whose gradients are not globally bounded, it seems difficult to show verticality, although such inequalities hold for ambient euclidean space [5,35].…”

“…Fischer-Colbrie and Schoen [23] studied stable minimal surfaces in three-dimensional manifolds with nonnegative scalar curvature and showed their rigidity. In our companion paper [21], we study minimal hypersurfaces in † and obtain existence and nonexistence results for area-minimizing hypersurfaces in such a †. Here, we restrict the dimension n C 1 4 for ambient product manifolds and investigate minimal graphs from the PDE point of view.…”