Stability of minimal surfaces and eigenvalues of the laplacian

Barbosa, João Lucas Marques; Carmo, Manfredo do

doi:10.1007/bf01215521

Cited by 56 publications

(33 citation statements)

References 14 publications

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“…Although D" may not be simply-connected, it is contained in a simply-connected domain that satisfies (2.21). By using Corollary (3.20) of [1] , we can still apply Ruchert's argument. Since he actually proves that J[; (O)(f) > 0, we arrive at a contradiction, and this proves our claim.…”

Section: S"mentioning

confidence: 98%

Stability of Hypersurfaces with Constant Mean Curvature

Barbosa

Carmo

2012

Manfredo P. Do Carmo – Selected Papers

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190

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Section: S"mentioning

confidence: 98%

Stability of Hypersurfaces with Constant Mean Curvature

Barbosa

Carmo

2012

Manfredo P. Do Carmo – Selected Papers

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130

190

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“…Barbosa and do Carmo generalized this result in several directions as e.g., for different target manifolds and for general dimensions [2].…”

Section: Introductionmentioning

confidence: 76%

A stability criterion for extremals of elliptic parametric functionals

Clarenz¹

2003

Pacific J. Math.

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In this paper we consider parametric integrals with elliptic integrands depending on the surface normal. The main result is a stability criterion for extremal immersions of those functionals, containing a result of Barbosa and do Carmo for minimal surfaces as a special case. Using similar techniques we are also able to show a condition for instability. The last section contains a simple proof of the fact that the surface normal of extremals of parametric integrals is topologically equivalent to a holomorphic function.

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“…, P N +3 of (see again Sect. 2 of [11] for an explanation of the notation X ( · , τ )) the proof of our stability theorem can be carried out following exactly the lines of the proof in [3]. A crucial reason for this is the following pointwise estimate of | KE | due to Heinz (see (3.3) in [7] or (26) in [11]) about each of the points e iτ 1 , .…”

Section: The Stability Theorem Of Barbosa and Do Carmomentioning

confidence: 97%

“…In this section, we prove a slightly modified variant of a stability theorem due to Barbosa and do Carmo (see [1] and [3], respectively) as stated below. In spite of the fact that the coefficient KE of the Schwarz operator A X , which is assigned to some fixed X ≡ X ( · , τ ) ∈ M i ( ), could behave rather "singularly" in those points e iτ 1 , .…”

Section: The Stability Theorem Of Barbosa and Do Carmomentioning

confidence: 98%