Existence theorems for minimal surfaces of nonzero genus spanning a contour

Tomi, Friedrich; Tromba, Anthony J.

doi:10.1090/memo/0382

Cited by 24 publications

(18 citation statements)

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“…When k = 1 and p = 0 then the theorem asserts the existence of an area minimizing disc which was treated in [21]. Theorem 1.2 generalizes the corresponding results in [15] and [33] from the setting of homogeneously regular Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 53%

“…In our first theorem we used the Douglas condition (1.1). A different condition, called condition of cohesion, was used by Shiffman [32], Courant [6], Tomi-Tromba [33]. In Theorem 8.2 we will show that if there exists an energy minimizing sequence satisfying the condition of cohesion then one can find an energy minimizer in Λ(M, Γ, X), even when X does not admit an isoperimetric inequality.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Moreover, we obtain continuity up to the boundary and interior Hölder regularity of solutions. Finally, we discuss the existence of energy minimizers under Courant's condition of cohesion used in [32], [6], [33].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Area minimizing surfaces of bounded genus in metric spaces

Fitzi¹,

Wenger²

2019

Preprint

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The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disctype surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 53%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Area minimizing surfaces of bounded genus in metric spaces

Fitzi¹,

Wenger²

2019

Preprint

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“…This procedure is well known and has been used very successfully by Douglas [6], Courant [5], Schoen and Yau [34], Sacks and Uhlenbeck [33], Tomi and Tromba [36] and others. This procedure is well known and has been used very successfully by Douglas [6], Courant [5], Schoen and Yau [34], Sacks and Uhlenbeck [33], Tomi and Tromba [36] and others.…”

Section: Statement and Discussion Of The Resultsmentioning

confidence: 99%

Comparison between second variation of area and second variation of energy of a minimal surface

Ejiri¹,

Micallef²

2008

Advances in Calculus of Variations

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We show that the difference between the Morse index of a closed minimal surface as a critical point of the area functional and its Morse index as a critical point of the energy is at most the real dimension of Teichmüller space. This enables us to bound the index of a closed minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold. Our method also yields surprisingly good upper bounds on the index of a minimal surface of finite total curvature in Euclidean space of any dimension.

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“…The fact that the completeness of M is irrelevant enables us to apply our result to solutions of the Plateau problem for arbitrary Jordan curves, whose existence is granted by R. Douglas in [14] (see also the treatises [11] and [12,13], as well as [19], [31]):…”

Section: Introductionmentioning

confidence: 99%

On the spectrum of bounded immersions

Bessa¹,

Jorge²,

Mari³

2015

J. Differential Geom.

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In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold ϕ : M m → N n and the Hausdorff dimension of its limit set lim ϕ. In particular, we prove that if ϕ :, where H Ψ is the generalized Hausdorff measure of order Ψ(t) = t 2 | log t|. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in R 3 , as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples.

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Existence theorems for minimal surfaces of nonzero genus spanning a contour

Cited by 24 publications

References 0 publications

Area minimizing surfaces of bounded genus in metric spaces

Area minimizing surfaces of bounded genus in metric spaces

Comparison between second variation of area and second variation of energy of a minimal surface

On the spectrum of bounded immersions

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