Constant scalar curvature metrics with isolated singularities

Abstract

We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on S N \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen [12], but the proof we give here, based on the techniques of [6], is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in [7] and [8]

Constant mean curvature surfaces with Delaunay ends

“…As usual, we also need this surjectivity with as good control as possible as the necksize shrinks. The results and proofs here are very close to those in [10].…”

“…On the other hand, the surfaces we construct have a rather different, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [10] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas' construction parallels the earlier construction of singular Yamabe metrics by Schoen [18].…”

Generalized connected sum construction for scalar flat metrics

“…They have been used to understand solutions to problems arising from the geometry (minimal and constant mean curvature surfaces [13,14], constant scalar curvature metrics [9,12,15], and recently even Einstein metrics [1]) and from the physic (Einstein constraint equations [7] and [8]). However most of the results are concerned with the connected sum at points (point-wise connected sum), whereas the case of connected sum along a submanifold (generalized connected sum or fiber sum) has received less attention.…”

Riemann Minimal Surfaces in Higher Dimensions

“…where the operators Q 2 and Q 3 enjoy the following property : There exists a constant c > 0 such that for all t ∈ R and for all w 1 , w 2 ∈ C 2,α ([t − 1, t + 1] × S n−1 ), we have 5) and, provided w 1 C 2,α + w 2 C 2,α ≤ 1, we also have…”

“…However, given the expansions of u ε,h ⊥ andū ε,ρ,h this is equivalent to solve To proceed with, we recall the following result [5] Using this result, the solvability of (4.2) reduces to a fixed point problem which can be written as (h,h) = S ε (h,h)…”