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Abstract

http://imrn.oxfordjournals.org/ Downloaded from 186 Sun-Yung A. Chang et al. Theorem 1.1.Let Ω ⊂ S n (n ≥ 5) be an open connected subset and let g S n be the standard metric on S n . Assume that there exists a metric g on Ω such that (Ω, g) is complete, g is conformal to g S n , |R| + |∇ g R| g ≤ c 0 , σ 1 (A) ≥ c 1 > 0, and σ 2 (A) ≥ 0, then dim(S n \Ω) < (n − 4)/2.On the other hand, we have the following theorem.Theorem 1.2. Let Ω ⊂ S n (n ≥ 3) be an open connected subset and let g S n be the standard metric on S n . Assume that there exists a metric g on Ω such that (Ω, g) is complete,In particular, this means that Ω = S n when n ≤ 4.If Q ≥ c 2 > 0 is replaced by Q ≥ 0, then when n ≥ 5, there exists dim(S n \Ω) ≤ (n − 4)/2 and when n = 3, there exists Ω = S 3 .We remark that if the condition |R| + |∇ g R| g ≤ c 0 in Theorem 1.1 is dropped, then one has dim(S n \Ω) ≤ (n − 4)/2. Similarly, if the condition |Rc| g + |∇ g R| g ≤ c 0 in Theorem 1.2 is dropped, then one has dim(S n \Ω) ≤ (n − 4)/2 when n ≥ 5, and Ω = S 3 when n = 3. These alternate versions follow from the proof presented below together with the coercivity of conformal factor proved in Section 8 and a method in [8] of using capacity theory to estimate the dimension of Λ, which replaces the use of Lemma 2.1.In general, the positivity of σ k (A) will yield a corresponding size control of Λ. Indeed, when k ≥ n/2, a calculation of [5] shows that when σ 1 (A) > 0, . . . , σ k (A) > 0, the Ricci tensor is positive definite, hence, the complement Λ is empty. For the intermediate range 3 ≤ k < n/2, the question is addressed in the forthcoming thesis of M. Gonzalez.It follows from the same consideration as in [8] that the following results hold. Corollary 1.3. Let (M n , g) (n ≥ 5) be a smooth compact locally conformally flat Riemannian manifold such that σ 1 (A) > 0 and σ 2 (A) ≥ 0, then π i (M) = 0 for any 2 ≤ i ≤ [n/2] + 1, and H j (M, R) = 0 for any n/2 − 1 ≤ j ≤ n/2 + 1.Corollary 1.4. Let (M n , g) (n ≥ 3) be a smooth compact locally conformally flat Riemannian manifold such that R > 0 and Q > 0, then when n ≥ 5, for any 2 ≤ i ≤ [n/2] + 1, π i (M) = 0; for any integer j with n/2 − 1 ≤ j ≤ n/2 + 1, H j (M, R) = 0 and when n = 3 or 4, the universal cover of M is conformal isomorphic to S n .We were recently informed that the result on the vanishing of cohomology groups in Corollary 1.3 is also derived in [4].In dimension four, one has the following slightly improved version of [2, Theorem 2].

Asymptotic Expansions of Solutions of the Yamabe Equation and the σk‐Yamabe Equation near Isolated Singular Points

“…Geometrically, for any positive solution u of the equation (1.1), the corresponding conformal metric g h u 4 n 2 jdxj 2 has a constant scalar curvature R g h n.n 1/.…”

“…We always assume that u has a nonremovable singularity at x h 0. Refer to [9] and its references for more information concerning the k -Yamabe equation, and in particular to [4], [6], and [7] for the size of the singular sets, and to [14] and [15] for the existence of solutions with isolated singularity.…”

Connected sums of closed Riemannian manifolds and fourth-order conformal invariants

“…Another testing ground is to consider closed locally conformally flat manifolds. Then the recent works in [Chang et al 2004] and [González 2005] indicate to us that the positivity of fourth-order curvature is indeed very informative about the topology of the underlying manifolds. We would also like to mention the work by Xu and Yang in [2001] where they demonstrated that positivity of the Paneitz-Branson operator is stable under the process of taking connected sums of two closed Riemannian manifolds.…”

“…The positivity of Paneitz invariant in dimension higher than 4 should be a topological constraint, as indicated by successful researches in [Chang and Yang 2002] (references therein) for a fourth-order analogue of how Gaussian curvature influences the geometry of surfaces in dimension 2. Another testing ground is to consider closed locally conformally flat manifolds.…”