The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature

Gauss curvature estimates for surfaces whose mean curvature is constant or has special properties

“…An important feature of this result is its independence of the distance of a point to the boundary, in contract with the results in Heinz [8], L. Simon [14], Choi-Schoen [4], Schoen-Simon-Yau [13], Anderson [1] and B. White [17].…”

“…4 and hence Theorem I.6 remain true for surfaces of prescribed mean curvature H with Δ(λ 4 |φ| 2 ) ≥ 0, i.e.…”

“…Theorem I. 4 Let M be an embedded (not necessarily simply-connected) surface in R 3 of constant mean curvature H. Set, for s ∈ R, s > 0,…”

Existence of minimal surfaces of arbitrarily large Morse index

“…Convergence theorems for minimal surfaces have been studied extensively in the past, especially in 3-dimension. Choi-Schoen [3] proved the compactness of embedded minimal surfaces of fixed topological type in a closed 3-manifold with positive Ricci curvature. Anderson [1] and White [33] respectively proved similar convergence and compactness results for minimal surfaces and surfaces that are stationary for parametric elliptic functionals in 3-manifold with bounded genus and bounded area.…”

On curvature estimates for constant mean curvature surfaces

“…Previous important examples of curvature estimates for constant mean curvature surfaces, assuming certain geometric conditions, can be found in the literature; see for instance [2,3,4,9,10,27,28,29,30,34,35].…”