Area‐minimizing projective planes in 3‐manifolds

Bray, Hubert L.; Brendle, Simon; Eichmair, Michael; Neves, André

doi:10.1002/cpa.20319

Cited by 44 publications

(72 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. Let P denote the space of projective planes embedded in M and denote The infimum in (10.1) is achieved by an embedded minimal projective plane Σ 0 (see for instance Proposition 5 in [4]). We will now use Σ 0 and a one-parameter min-max argument to produce a second minimal projective plane.…”

Section: Minimal Embedded Projective Planes In Rpmentioning

confidence: 99%

Minimal 2-spheres in 3-spheres

Haslhofer¹,

Ketover²

2019

Duke Math. J.

View full text Add to dashboard Cite

We prove that any manifold diffeomorphic to S 3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal twosphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three manifolds. We apply our methods to solve a problem posed by S.T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered about the origin in R 4 . Finally, considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp.

show abstract

Section: Minimal Embedded Projective Planes In Rpmentioning

confidence: 99%

Minimal 2-spheres in 3-spheres

Haslhofer¹,

Ketover²

2019

Duke Math. J.

View full text Add to dashboard Cite

show abstract

“…Imposing a curvature condition on (M, g) and perhaps some extra condition on Σ is possible to obtain area estimates and rigidity results for Σ and (M, g). For example, assuming a lower bound on the scalar curvature and that the surface is minimizing or does have index 1 with respect to the functional Area − H · V olume, some rigidity results were obtained in [7,26,5,6,22,16]. There are also related results for free boundary minimal surfaces in 3-manifolds with boundary, see [3,19], and for MOTS in a spacetime, see [12,20].…”

Section: Introductionmentioning

confidence: 97%

Area estimates and rigidity of non-compact 𝐻-surfaces in 3-manifolds

Lima

2019

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

For appropriately values of H, we obtain an area estimate for a complete non-compact H-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighbourhood of the mean convex side of the surface must be isometric to a hyperbolic Fuchsian manifold. Also, we show by an counterexample that although that area estimate holds for minimal surfaces, one does not have rigidity for equality in this case.

show abstract

“…The MinA condition is natural in this setting given the crucial role that stable minimal surfaces played in Schoen-Yau's proof of the torus rigidity theorem [SY79]. Also, the MinA condition has appeared in the rigidity results of Bray, Brendle and Neves [BBN10] for area minimizing 2-spheres in 3-spheres and Bray, Brendle, Eichmair and Neves [BBEN10] for area minimizing projective planes in 3-manifolds.…”

Section: Introductionmentioning

confidence: 99%