Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary

By A. Fraser, M. Li. 2012

151 citations recorded by the source corpus.

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies apriori area and curvature estimates for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness result for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/43201b27a9bcf713051898b8fd2ddec1c857dd2c

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Free boundary minimal surfaces of unbounded genus (2016, background): “) orthogonally (if at all). Remark 3.3. The connectedness of Γ follows from the Frankel-type property of embedded free boundary minimal surfaces in convex bodies: any two must intersect (Lemma 2.4 in [FL]). Remark 3.4. By “surgeries” inii) is meant either “G-equivariant neckpinch” (cf. Remark 1.6 in”
Sharp eigenvalue bounds and minimal surfaces in the ball (2012, methodology): “raphs over the tangent plane. By Allard-type minimal surface regularity theorems (see [GJ]), a boundary point xis a smooth point if and only if the density at xis equal to 1/2. We use the argument of [FL] to ﬁrst show that Σ∞ is connected. Indeed if we have two connected components Σ′ and Σ′′, then ”
Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc (2017, methodology): “ary since their properties are greatly aected by the ambient geometry. For example, Fraser [14] [13] used index estimates to study the topology of Euclidean domains with k-convex boundary. Fraser-Li [8] proved a smooth compactness result for embedded free boundary minimal surfaces when the ambient m”
Compactness analysis for free boundary minimal hypersurfaces (2017, background): “The starting point of our discussion is the following result by Fraser and Li [14], which should be regarded as a free boundary analogue of the classic theorem by Choi and Schoen [8]:”
Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory (2016, methodology): “of . (6)The boundary homeomorphism @: I(m;j) !I(m;j) is given by @( 1 m) := Xm i=1 ( 1)˙(i) 1 @ i m; where ˙(i) = P l<i dim( l), @[a;b] = [b] [a] if [a;b] 2I(1;j) 1, and @[a] = 0 if [a] 2I(1;j) 0. (7)The distance function d : I(m;j) 0 I(m;j) 0 !N on vertices is dened as d(x;y) := 3j Xm i=1 jx i y”
Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, result): “In [23], we proved a lower bound of σ1(Σ) which is similar to the bound obtained by Choi and Wang [11] for closed embedded minimal hypersurface in S.”
Free boundary minimal surfaces in the unit 3-ball (2015, background): “Li conjectured that it is the only free boundary minimal surface of topological annulus type [1].”
A Characterization of the Critical Catenoid (2016, methodology): “e unit ball B3 are ﬂat equators. It is not known whether or not similar rigidity theorems hold for surfaces with larger genus or number of boundary components, but in this direction and Fraser and Li [7] made the following conjecture: Conjecture 1 (Fraser and Li, [7]). Up to congruences, the critica”