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

By N. Ejiri, Mario J. Micallef. 2007

50 citations recorded by the source corpus.

Abstract 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2880964e7ba5170f890e024a73489a728c721e5b

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In Applied Mathematics

1. Christos Mantoulidis
2. Otis Chodosh
3. Antoine Song

In Geometry and Topology

1. Christos Mantoulidis
2. Otis Chodosh

Citations in the corpus

• Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates (2018, background): “(Σp) ≤ Cp 1 3 . (6.1) Because p= PN m=1 ind(Σ (m) p ), (6.1) implies that ind(Σm p ) ≥ Cp 2 3 for some m∈ {1,...,N}. For this particular m, areag(Σ (m) p ) ≤ Cp 1 3 and the estimate of Ejiri–Micallef [EM08] used above implies that genus(Σ(m) p ) ≥ C −1 ind(Σ(m) p ) ≥ C −1p2 3 . This completes the pr”