Stable constant mean curvature tori and the isoperimetric problem in three space forms

Ritoré, Manuel; Ros, Antonio

doi:10.1007/bf02566501

Cited by 61 publications

(69 citation statements)

References 20 publications

(32 reference statements)

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“…M = S 1 × R 2 ), the theorem allows us to conclude that g = 0 or 1 and therefore Σ is either a round sphere or a right cylinder, as follows from Ritoré and Ros [20]. We remark that this was previously known only in the embedded case.…”

mentioning

confidence: 58%

“…Thus, it follows from (1) that Σ is totally geodesic. If Σ is connected, then Ros [25] proves that the genus of Σ is 3 (for earlier partial results see [6,20,19,30]). …”

Section: Preliminariesmentioning

confidence: 99%

“…If Σ is singly periodic (k = 1), Aleksandrov's technique gives that it must be a Delaunay surface of revolution. If moreover the quotient surface Σ/Γ ⊂ R 3 /Γ is stable, then Ritoré and Ros [20] prove that Σ is a right circular cylinder. Doubly periodic constant mean curvature surfaces (k = 2) were first constructed by Lawson [11].…”

Section: Preliminariesmentioning

confidence: 99%

“…In the case g = 1, Ritoré and Ros [20] proved that stable surfaces are quotients of right circular cylinders. If g = 2, then the connected components of the preimage Σ ⊂ R 3 of Σ are at most doubly periodic.…”

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confidence: 99%

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Stable periodic constant mean curvature surfaces and mesoscopic phase separation

Ros

2007

Interfaces Free Bound.

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We give a first comprehensive description of the stable solutions of the periodic isoperimetric problem in the case of lattice symmetry. This result is intended to elucidate the geometry of certain sophisticated interfaces appearing in mesoscale phase separation phenomena. We prove that closed, stable, constant mean curvature surfaces in R 3 /Γ , Γ ⊂ R 3 being a discrete subgroup of translations with rank k, have genus k. Finally, we extend the genus estimate to ambient spaces of the type M × R, where M is a nonnegatively curved surface.

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mentioning

confidence: 58%

“…Thus, it follows from (1) that Σ is totally geodesic. If Σ is connected, then Ros [25] proves that the genus of Σ is 3 (for earlier partial results see [6,20,19,30]). …”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

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Stable periodic constant mean curvature surfaces and mesoscopic phase separation

Ros

2007

Interfaces Free Bound.

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“…Previously do Carmo and Peng [6] and Fischer-Colbrie and Schoen [8] had shown that the only stable (index zero) complete minimal surface is the plane. From the work by Ritoré and Ros [14] a classification of index one minimal surfaces in P 3 can be obtained: it must be a two-fold covering of a linear subvariety or a tube of certain radius around a line. These authors [15] also obtained a compactness result for the space of index one minimal surfaces in flat three tori.…”

Section: Introductionmentioning

confidence: 99%

Compact minimal hypersurfaces with index one in the real projective space

Carmo¹,

Ritoré²,

Ros³

2000

Comment. Math. Helv.

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Abstract. Let M n be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S n+1 is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.We prove that if M is the real projective space P n+1 = S n+1 /{±}, obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface S n 1 (R 1 ) × S n 2 (R 2 ) ⊂ S n+1 obtained as the product of two spheres of dimensions n 1 , n 2 and radius R 1 , R 2 , with n 1 + n 2 = n, R 2 1 + R 2 2 = 1 and n 1 R 2 2 = n 2 R 2 1 .

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Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

Chodosh

Eichmair

Shi

et al. 2021

Comm Pure Appl Math

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Let (M, g) be an asymptotically flat Riemannian 3‐manifold with nonnegative scalar curvature and positive mass. We show that each leaf of the canonical foliation of the end of (M, g) through stable constant mean curvature spheres encloses more volume than any other surface of the same area. Unlike all previous characterizations of large solutions of the isoperimetric problem, we need no asymptotic symmetry assumptions beyond the optimal conditions for the positive mass theorem. This generality includes examples where global uniqueness of the leaves of the canonical foliation as stable constant mean curvature spheres fails dramatically. Our results here resolve a question of G. Huisken on the isoperimetric content of the positive mass theorem. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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Stable constant mean curvature tori and the isoperimetric problem in three space forms

Cited by 61 publications

References 20 publications

Stable periodic constant mean curvature surfaces and mesoscopic phase separation

Stable periodic constant mean curvature surfaces and mesoscopic phase separation

Compact minimal hypersurfaces with index one in the real projective space

Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

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