A mountain pass theorem for minimal hypersurfaces with fixed boundary
- PDF / 491,690 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 20 Downloads / 207 Views
Calculus of Variations
A mountain pass theorem for minimal hypersurfaces with fixed boundary Rafael Montezuma1,2 Received: 30 April 2018 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold γ , of mountain pass type, contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound γ . In order to do so, we develop min–max methods similar to those of De Lellis and Ramic (Ann. Inst. Fourier 68(5): 1909 –1986, 2018) adapted to the discrete setting of Almgren and Pitts. Our approach allows one to consider the case in which the two stable hypersurfaces with boundary γ intersect at interior points. Mathematics Subject Classification 53C20 · 58E05 · 58E12
1 Introduction We are concerned with the problem of existence of a third embedded minimal surface spanning a smooth closed curve γ , possibly with multiple components, when it is known a priori the existence of two other minimal surfaces of minimum type with boundary γ , or satisfying other natural local minimization condition. We are also interested in the higher dimensional codimension one version of this problem in the Riemannian setting. Since the classical works of Morse and Tompkins [17], and Shiffman [21], in the parametric setting of the Plateau’s problem, this type of problem has been studied extensively in the case of 2-dimensional submanifolds spanning a contour in Euclidean space. See, for instance, the deep results contained in [12] and in the references therein. In a recent paper, [6], De Lellis and Ramic obtained a very general result of that type. Indeed, our work was motivated by a question posed in [6]. We also apply tools and ideas from that work. Our main result is the following.
Communicated by A.Neves.
B
Rafael Montezuma [email protected]
1
Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA
2
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, CE 60455-760, Brazil 0123456789().: V,-vol
123
188
Page 2 of 30
R. Montezuma
Theorem 1.1 Let M n+1 be a compact, oriented, Riemannian manifold with strictly convex boundary, and γ n−1 be a closed, embedded, oriented, smooth submanifold of ∂ M. Suppose that there exist distinct embedded, oriented, smooth, strictly stable minimal hypersurfaces 1 and 2 , such that ∂i = γ , for i = 1, 2, and 1 and 2 are homologous. Suppose also that all connected components of each i have non-empty boundary. Then, there exists a distinct embedded minimal hypersurface in M, which has a singular set sing() = \ of Hausdorff dimension at most n − 7, and sing() ∩ (∂ M) = ∅. Moreover, ∂ = γ and there is a connected component of which is contained neither in 1 , nor in 2 . The minimal hypersurface obtained in the above theorem could be the union of an embedded closed minimal hypersurface that
Data Loading...
