Splitting of 3-Manifolds and Rigidity of Area-Minimising Surfaces

Free Boundary Hypersurfaces with Non-positive Yamabe Invariant in Mean Convex Manifolds

“…For surfaces of genus g(Σ) > 1, Nunes [23,Theorem 3] has obtained an interesting rigidity result for minimal hyperbolic surfaces in three-manifolds with scalar curvature bounded by a negative constant. There is also an unified point of view with alternative proofs about these cases considered by Micallef and Moraru [20]. For a good reference about other rigidity theorems we refer the reader to [5].…”

“…It follows from Lemma 1 that, since the lapse function satisfies the homogeneous Neumann problem, ℓ t is constant (as function of t) at each Σ t . The function µ(t, x) = 0 and the vector field N t is parallel for all (t, x) ∈ (−ε, ε) × Σ (see [20] or [23]) and its flow is the exponential map, i.e., f(t, x) = exp x (tN (x)) ∀x ∈ Σ which is an isometry for all t ∈ (−ε, ε). Hence, the metric of M near Σ must split as dt 2 + g.…”

Free boundary stable hypersurfaces in manifolds with density and rigidity results

“…We will obtain topological and geometrical restrictions for strongly f -stable hypersurfaces under certain curvature and boundary assumptions on the ambient manifold. Our statements and proofs are inspired by previous results for the Riemannian case, see [7], [5], [27], [24], [11], [1], and for hypersurfaces with empty boundary in manifolds with density, see [16], [23] and [15].…”

“…The simplest examples of strongly f -stable totally geodesic hypersurfaces satisfying Ric f (N, N ) = 0 and II(N, N ) = 0 are the horizontal slices {s} × Σ in a Riemannian product R × Σ, where Σ is a compact Riemannian manifold of non-negative Ricci curvature, and the logarithm of the density f is a linear function in R. These are not the unique examples we may give, i.e., the existence of compact strongly f -stable hypersurfaces in the above conditions does not imply that the metric of M splits, even locally, as a product metric, see [24]. However, in Theorem 4.2 we prove the following rigidity result:…”

Disks area-minimizing in mean convex Riemannian $n$-manifolds

“…Moreover, if the equality holds then the universal cover of (M, g) is isometric to the standard cylinder S 2 × R up to scaling. For more results concerning to rigidity of 3-dimensional closed manifolds coming from area-minimising surfaces, see [3], [4], [5], [6], [8]. In [10], J. Zhou showed a version of Bray, Brendle and Neves [2] result for high co-dimension: for n + 2 ≤ 7, let (M n+2 , g) be an oriented closed Riemannian manifold with R g ≥ 2, which admits a non-zero degree map F : M → S 2 × T n .…”