Asymptotically flat three-manifolds contain minimal planes

Chodosh, Otis; Ketover, Daniel

doi:10.1016/j.aim.2018.08.010

Cited by 8 publications

(9 citation statements)

References 38 publications

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“…In order to analyze the system determined by equations ( 7), (8), and (9), we consider the following separation of variables…”

Section: Separation Of Variables and Reduction To A Ricatti Equationmentioning

confidence: 99%

“…On the other hand, Chodosh and Ketover [8] proved existence of many complete properly embedded minimal planes in asymptotically flat threemanifolds containing no closed embedded minimal surfaces. This was later generalized by Mazet and Rosenberg [15].…”

Section: Introductionmentioning

confidence: 99%

“…This was later generalized by Mazet and Rosenberg [15]. In the work [8], the authors apply degree theoretic methods, what makes it hard to give explicit estimates on the Morse index of such planes. In the same paper, they suggest also the naturally related search for unbounded free-boundary minimal surfaces in asymptotically flat manifolds with closed minimal boundaries, and the study of such surfaces in the exact Schwarzschild metric.…”

Section: Introductionmentioning

confidence: 99%

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On free boundary minimal surfaces in the Riemannian Schwarzschild manifold

Montezuma¹

2019

Preprint

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Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat 3-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manifold.More precisely, we observe that the simplest minimal surface in this space has Morse index one. We prove also a relationship between the length of the boundary and the density at infinity of general minimal surfaces satisfying a free-boundary condition along the horizon.

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“…In order to analyze the system determined by equations ( 7), (8), and (9), we consider the following separation of variables…”

Section: Separation Of Variables and Reduction To A Ricatti Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On free boundary minimal surfaces in the Riemannian Schwarzschild manifold

Montezuma¹

2019

Preprint

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“…The existence of minimal surfaces in hyperbolic 3-manifolds has been proved by Collin-Hauswirth-Mazet-Rosenberg [CHMR17], Huang-Wang [HW17] and Coskunuzer [Cos18]. In [CK18], Chodosh and Ketover proved the existence of minimal planes in asymptotically flat 3-manifolds. In [CL20], Chambers and Liokumovich proved that every complete Riemannian manifold with finite volume contains a complete minimal hypersurface with finite area.…”

Section: Introductionmentioning

confidence: 98%

The Allen-Cahn equation on the complete Riemannian manifolds of finite volume

Dey

2021

Preprint

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The semi-linear, elliptic PDE AC ε (u) := −ε 2 ∆u + W ′ (u) = 0 is called the Allen-Cahn equation. In this article we will prove the existence of finite energy solution to the Allen-Cahn equation on certain complete, non-compact manifolds. More precisely, suppose M n+1 (with n + 1 ≥ 3) is a complete Riemannian manifold of finite volume. Then there exists ε 0 > 0, depending on the ambient Riemannian metric, such that for all 0 < ε ≤ ε 0 , there existsOur result is motivated by the theorem of Chambers-Liokumovich [CL20] and Song [Son19], which says that M contains a complete minimal hypersurface Σ with 0 < H n (Σ) < ∞. This theorem can be recovered from our result.

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“…For the 3-dimensional Riemannnian Schwarzschild space M 3 1 , O. Chodosh and D. Ketover [6] proposed to study the Morse index of the annuli surface Σ 0 , i.e., the maximum number of directions, tangential along ∂M , in whose the surface can be deformed in such a way that its area decreases, and also to investigate the existence of unbounded non-totally geodesic free boundary minimal surfaces. It follows from the work of A. Carlotto [3] that the Morse index of Σ 0 can not be zero.…”

Section: Introductionmentioning

confidence: 99%