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

Ejiri, Norio; Micallef, Mario

doi:10.1515/acv.2008.009

Cited by 36 publications

(43 citation statements)

References 31 publications

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“…In particular, Corollary 1.3 and the results from [7,3] show that in the case of bounded area, a bound on the Morse index is (qualitatively) equivalent to a bound on the total curvature.…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

“…There are many known relationships between the index of minimal submanifolds and their topological and analytic properties. In particular, given an immersed minimal submanifold in a closed Riemannian manifold, one knows that the Morse index is bounded linearly from above in terms of area and total curvature, see Ejiri-Micallef [7] for n = 2 and Cheng-Tysk [3] for n ≥ 3. Moreover, for properly immersed minimal hypersurfaces in Euclidean space, we have the same result -except this time the upper bound is purely in terms of the total curvature (and thus implicitly contains information on the number of ends the minimal hypersurface has).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Buzano

Sharp

2018

Trans. Amer. Math. Soc.

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We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit hypersurface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space -all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.

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“…In particular, Corollary 1.3 and the results from [7,3] show that in the case of bounded area, a bound on the Morse index is (qualitatively) equivalent to a bound on the total curvature.…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Buzano

Sharp

2018

Trans. Amer. Math. Soc.

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“…Finally, we explain how our main results imply a qualitative version of the fact that finite index is equivalent to finite total curvature. Theorems 1.1 and 1.3, combined with the Jorge-Meeks formula yield the following result (the upper bound is due 5 to Ejiri-Micallef [EM08]).…”

Section: Introductionmentioning

confidence: 87%

On the topology and index of minimal surfaces

Chodosh¹,

Máximo

2016

J. Differential Geom.

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For an immersed minimal surface in R 3 , we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index.Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there is no complete two-sided immersed minimal surface in R 3 of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.

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“…In the uniqueness proof it is necessary to show that the energy has high enough index, so in order to do this we solve a Cauchy-Riemann equation on to add a tangential component to certain normal variations to make them conformal. We note that a systematic study of the relationship between second variation of energy and area was done by Ejiri and Micallef [5].…”

Section: Theorem 12mentioning

confidence: 94%

Sharp eigenvalue bounds and minimal surfaces in the ball

2015

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We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.Comment: 52 pages. Final version that appeared in Invent. Math.: Presentation of Proposition 4.3 improved, other minor edits throughou

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

Cited by 36 publications

References 31 publications

Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

On the topology and index of minimal surfaces

Sharp eigenvalue bounds and minimal surfaces in the ball

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