Isolated singularities of solutions to the Yamabe equation in higher dimensions II

Introduction

Let $g_0$ be the standard quotient metric on $\mathbb{RP}^{n}$, induced by the round metric on $S^{n}(1/2)$ under the antipodal identification. For a Riemannian metric $g$, let $\operatorname{Scal}_{g}$ denote its scalar curvature, and define the conformal Laplacian by

We write $d_g$ for the distance induced by $g$.

The main result of this paper is the following.

We make the following remarks.

• In Part I [11], we completed the classification of isolated singularities of positive solutions to the Yamabe equation in punctured balls, for arbitrary smooth Riemannian metrics in dimensions $3 \le n \le29$: every such singularity is either removable or the solution is asymptotic to a Fowler solution—a radial model singular solution in the flat metric on the Euclidean space with origin as its singular point. See also Marques [15] in dimensions $3 \le n \le5$ and Xiong–Zhang [22] in dimension 6. By (3), the above theorem shows that this classification cannot be extended to dimensions 30 and higher.

• We will see in the proof of the main theorem that, for any $\tau> 0$,

in local coordinates centered at $o$. This shows that the upper bound assumption in Han–Xiong–Zhang [12 Theorems 1.2 and 1.3] is necessary when $n \ge30$.

• By (2) and (1), the conformal metric $u_{\infty}^{4/(n-2)}g_{\infty}$ is complete and has scalar curvature $4n(n-1)$. To the best of our knowledge, this is the first example of a complete conformal metric of constant positive scalar curvature whose conformal factor, relative to a smooth background metric, grows faster than the Fowler rate along a sequence approaching an isolated singularity. See Schoen [21], Mazzeo–Smale [18], Mazzeo–Pollack–Uhlenbeck [17], Mazzeo–Pacard [16], Byde [5], Santos [20], and others for the singular Yamabe problem. In a forthcoming paper, we prove that if $(M,g)$ is a smooth, connected, closed Riemannian manifold of dimension $n \ge3$, has positive Yamabe constant, and is not conformally diffeomorphic to the round sphere, then, for every nonempty finite set $\{P_1,\ldots,P_m\} \subset M$, there exists a positive smooth function $u$ on $M \setminus\{P_1,\ldots,P_m\}$ such that $u^{4/(n-2)}g$ is complete and has constant positive scalar curvature, and $u$ has Fowler asymptotics at each $P_i$.

• For a fixed positive Yamabe conformal class on a closed manifold not conformally diffeomorphic to the round sphere, Khuri–Marques–Schoen [13] proved compactness through dimension 24, using the positive mass theorem in the final global argument. In dimensions $n \ge52$, Brendle [3] constructed smooth metrics on the sphere with a noncompact sequence of smooth solutions, developing exactly one bubble; Brendle–Marques [4] extended the construction to $25 \le n \le51$. For every $n \ge25$, Gong–Li [9] constructed sequences of positive solutions on the sphere for a fixed smooth background metric, with both the number of bubbles and the volumes of the associated conformal metrics tending to infinity. In their construction, the bubble centers approach a single point, and the conformal factors converge to zero locally uniformly away from that point.

Overview of the proofs

Our construction proceeds by an iterative Lyapunov–Schmidt reduction, starting from a Fowler solution on the punctured round projective space. At each stage, we perturb the metric locally and construct a new exact solution with an additional bubble, while retaining Fowler asymptotics sufficiently close to the puncture. The insertion points approach the puncture. The metrics converge smoothly on the whole manifold, and the solutions converge smoothly on compact subsets away from the puncture. The limiting solution retains the peaks created at successive stages and violates the Fowler upper bound. Its behavior therefore differs from the Fowler asymptotics of the isolated ends in the constructions of Schoen [21] and Mazzeo–Pacard [16]. To accommodate the infinite energy of the Fowler background, we introduce a renormalized relative energy for the finite-dimensional reduction. We adapt the localized metric perturbation constructions of Brendle [3] and Brendle–Marques [4] to successive backgrounds with Fowler asymptotics, preserving the nondegeneracy needed to continue the iteration. For a related infinite-dimensional gluing construction in the fractional Yamabe setting with the flat metric, we refer to Ao–DeLaTorre–González–Wei [1].

We now sketch our construction.

Let $\gamma_0 = |dx|^2$ on $B_1(0)\setminus\{0\}$, with the antipodal boundary matching on $\partial B_1(0)$, and take a fixed Fowler solution $V_0(x) = |x|^{-(n-2)/2}v_*(-\log|x|)$ as the initial solution for this flat background, where $v_*$ is a fixed non-constant periodic even positive Fowler profile chosen in Section 2 with minimum value $m_*$ (called the neck size) and is normalized so that $v_*(0) = m_*$. The initial conformal metric is

At stage $j$, after fixing a new insertion ball, choose $\mu_j > 0$ sufficiently small. Proposition 2.2 perturbs $\gamma_{j-1}$ in that ball to a new metric $\gamma_j$ and adds a bubble to the current solution $V_{j-1}$; the Lyapunov–Schmidt correction makes the resulting $V_j$ an exact positive solution for the perturbed metric $\gamma_j$. Thus the conformal metric after stage $j$ is

and $(\gamma_j,V_j)$ is the input for the next stage.

Each $V_j$ has the same Fowler profile and neck parameter at the puncture as the initial solution; its periodic factor may be translated by a phase. In the Euclidean coordinates $x$, the end condition has the form

where the remainder has the weighted derivative bounds specified in (15). Proposition 2.2 gives small metric and solution changes outside the insertion ball and a high peak inside it.

Proposition 2.2 also proves that the augmented linearization in (16) is an isomorphism. Its unknown consists of a scalar phase increment and a weighted function correction; the scalar changes the Fowler phase near the puncture. This is the inverse estimate needed for the next insertion.

The Fowler end has infinite energy, so we use a finite relative energy to measure the change produced by inserting a bubble. With the insertion scale fixed, the Lyapunov–Schmidt reduction first solves for the phase increment and the function correction, and then determines the bubble parameters by finding a critical point of the reduced energy. The main difficulty is to show that neither the interaction between the bubble and the Fowler background nor the change in energy caused by adjusting the Fowler phase and the function through the projected equation destroys the nondegenerate minimum of the leading energy. The dimension assumption $n \ge30$ provides the necessary order separation. At the resulting critical point, the remaining finite-dimensional error vanishes, giving an exact solution. Sections 3–5 give the construction and estimates.

We choose each insertion scale after fixing the current background. The metric increments are summable in every smooth norm, and the solution increments are summable outside the new ball. A local comparison transfers the boundary lower bound into that ball. Consequently, the limiting solution satisfies the lower bound (2), which gives completeness. The peak condition (30), together with the summability of later solution increments, yields (3).

Section 2 defines the class of backgrounds, states the insertion proposition, and deduces Theorem 1.1. Section 3 proves the initial inverse estimate and develops the nonlinear phase correction and the relative energy on punctured balls. Section 4 constructs the metric perturbation and solves the projected equation after adding a bubble. Section 5 chooses the bubble parameters by finding a nondegenerate critical point of the reduced energy. Section 6 proves the insertion proposition, including invertibility of the full linearization at the new solution. Appendix § 7.§ 1 proves the nondegeneracy of the leading bubble energy. Subsection A.2 establishes a local solution theorem on arbitrary smooth backgrounds, and Subsection A.3 gives its second-order asymptotics; these local results hold in every dimension $n \ge3$. Subsection A.4 proves the regularity and weighted-convergence results.

Constants and uniformity. We fix $n \ge30$ in Sections 2–6. The constant $C > 0$ in the estimates may change from line to line. During one insertion, it may depend on the fixed dimension, Fowler profile, background, weight, insertion point, radius, and cutoffs. It is independent of $\mu$ and uniform on the stated compact parameter set. The inverse norm at the new solution may depend on the chosen insertion scale.

Bubble insertion and the proof of the main theorem

In this section, we use the insertion result and a local lower bound to prove Theorem 1.1. Lemma 2.1 supplies the initial background, Proposition 2.2 preserves its properties at each insertion, and Lemma 2.4 provides the lower bound needed for completeness.

We first describe the punctured ball and its boundary identification. Put $X = B_1(0) \setminus\{0\} \subset\mathbb{R}^n$. We use fixed Euclidean coordinates $x$, with $r = |x|$, throughout the construction. In these coordinates, the Euclidean metric $\gamma_0$ has components $(\gamma_0)_{ij} = \delta_{ij}$. The closed ball with antipodal boundary points identified represents $\mathbb{RP}^n$; its center represents $o$. The round metric of radius $1/2$ and its Euclidean conformal factor are

Our analytic unknowns are the Euclidean representatives

Thus $g = \omega^{4/(n-2)}\gamma$ and $u = \omega^{-1}V$. In particular, $u^{4/(n-2)}g = V^{4/(n-2)}\gamma$. The conformal covariance formula is

Under the change (2.2), the Yamabe equation becomes $-L_\gamma V = n(n - 2)V^p$ on $X$, where $p := \frac{n+2}{n-2}$. All metrics $\gamma$ used at a finite stage are smooth on $\overline{B}_1$, equal to $\gamma_0$ near $0$ and on a fixed collar of $\partial B_1$, and satisfy $dV_\gamma= dx$.

The antipodal identification is expressed by the Kelvin extension

At $r = 1$,

Matching values and radial derivatives therefore gives