• Corpus ID: 231846894

Area, Scalar Curvature, and Hyperbolic 3-Manifolds

@inproceedings{Lowe2021AreaSC,
  title={Area, Scalar Curvature, and Hyperbolic 3-Manifolds},
  author={Ben Lowe},
  year={2021}
}
  • Ben Lowe
  • Published 6 February 2021
  • Mathematics
Let M be a closed hyperbolic 3-manifold that contains a closed immersed totally geodesic surface Σ. Then if g is a Riemannian metric on M with scalar curvature greater than or equal to −6, we show that the area in g of any surface homotopic to Σ is greater than or equal to the area of Σ in the hyperbolic metric, with equality only if g is isometric to the hyperbolic metric. We also consider a functional introduced by Calegari-MarquesNeves that is defined by an asymptotic count of minimal… 
MINIMAL SURFACE ENTROPY OF NEGATIVELY CURVED MANIFOLDS
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References

SHOWING 1-10 OF 33 REFERENCES
Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds
We prove that if M is a three-manifold with scalar curvature greater than or equal to −2 and Σ⊂M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally
Area‐minimizing projective planes in 3‐manifolds
Let (M,g) be a compact Riemannian manifold of dimension 3, and let ℱ denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$. We study the infimum of the
Deforming metrics in the direction of their Ricci tensors
In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive
Volumes of balls in large Riemannian manifolds
We prove two lower bounds for the volumes of balls in a Riemannian manifold. If (M n ;g) is a complete Riemannian manifold with lling radius at least R, then it contains a ball of radius R and volume
Minimal discs in hyperbolic space bounded by a quasicircle at infinity
We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm
Embeddedness of least area minimal hypersurfaces
  • A. Song
  • Mathematics
    Journal of Differential Geometry
  • 2018
E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this
On Area Comparison and Rigidity Involving the Scalar Curvature
We prove a splitting theorem for Riemannian $$n$$n-manifolds with scalar curvature bounded below and containing certain area-minimizing hypersurfaces (Theorem 3). The theorem will follow from an area
Asymptotic stability of the cross curvature flow at a hyperbolic metric
We show that for any hyperbolic metric on a closed 3-manifold, there exists a neighborhood such that every solution of a normalized cross curvature flow with initial data in this neighborhood exists
The entropy formula for the Ricci flow and its geometric applications
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric
Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces
In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics
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