Morse theory for the area functional

Marques, Fernando C.; Neves, André

doi:10.1007/s40863-019-00159-y

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Related Backgrounds On Cmc and Minimal Surfaces1

Introduction1

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2021

2023

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Cited by 5 publications

(2 citation statements)

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“…When combined with [36], this implies that, for bumpy metrics, there exists a closed minimal hypersurface of Morse index 𝑝 for each 𝑝 ∈ ℕ. Recently, the Morse inequalities for the area functional were proved for bumpy metrics by Marques et al [32].…”

Section: Related Backgrounds On Cmc and Minimal Surfacesmentioning

confidence: 99%

See 1 more Smart Citation

Existence of constant mean curvature 2‐Spheres in Riemannian 3‐spheres

Cheng

Zhou

2023

Comm Pure Appl Math

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We prove the existence of branched immersed constant mean curvature (CMC) 2‐spheres in an arbitrary Riemannian 3‐sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3‐sphere is positively curved. To achieve this, we develop a min‐max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi‐harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min‐max values to gain energy upper bounds for min‐max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.

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Section: Related Backgrounds On Cmc and Minimal Surfacesmentioning

confidence: 99%

“…Recently, the Morse inequalities for the area functional were proved for bumpy metrics by Marques et al. [32].…”

Section: Introductionmentioning

confidence: 99%