In this section we discuss a number of sharp Sobolev trace inequalities involving the W3,2(X)-seminorm of a function when dimX>6. First we consider a norm inequality which establishes the trace embedding
In this section we describe some examples to which Theorem 1.1 applies, focusing on the fully nonlinear case k≥2. Together these examples prove Theorem 1.2, Theorem 1.3, and Theorem 1.4. From the perspective of the boundary, our examples are conform…
We prove Theorem A in the case n=4; proofs of the other cases follow analogously. Let M=R×C4 and set
We conclude by discussing examples of CVIs through two families. First, in Subsection 8.1 we describe Branson’s Q-curvatures [5, 26] and the renormalized volume coefficients  from the point of view of CVIs, noting in particular that they provide…
The goal of this section is to give a partial classification of metrics g∈¯¯¯¯¯¯¯Γ+k in the conformal class of the round metric on the upper hemisphere Sn+1+ for which σgk≡0 and Hgk is constant along the boundary Sn=∂Sn+1+. These results generalize …
In this section we compute the linearizations of the total weighted σ1- and σ2-curvatures with the goal of showing that weighted Einstein manifolds are among their critical points. Specifically, in the case of scale zero, we show that quasi-Einstein…
In order to prove Theorem 1.2, we first compute the second variation of the V3-functional at a volume-normalized shrinking gradient Ricci soliton.
The proof of Theorem 1.1 proceeds analogously to the proof of the corresponding result on four-dimensional Riemannian manifolds  with one important difference: P′ is defined as a C∞(M)/P⊥-valued operator; in particular, it is a nonlocal operator.…
In this section, we will prove Theorem 1.3. First, we can repeat the proof of Theorem 8.1 with minor change and get the following result.
Consider now the special case of a Riemannian four-manifold (X4,g) with boundary M3=∂X. In this case the boundary operators B3k:C∞(X)→C∞(M) from Theorem 1.1 are given by
As an application of Theorem 5.4, we prove the following sharp Sobolev trace inequality for γ-admissible compactifications of hyperbolic space with γ∈(1,2). This statement is more general than Theorem 1.3 in that it involves the full trace on Cγ and…
As discussed in the introduction, a calculation from  and stability results from [2, 13] show that real ellipsoids in C2, viewed as deformations of the standard CR three-sphere, satisfy conditions (1), (3), (4) and (5) of Theorem 1.3. The continu…
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