A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $σ_k$-curvature Metrics

Abstract: Let $k\in\mathbb N$ satisfy $1<k<n/2$, and let $Σ^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\setminusΣ$ satisfying $λ(g^{-1}A_g)\in\Gk$ and $$σ_k(g^{-1}A_g)=κ\&gt;0$$ must obey [ p\leq p_k(n), ] where $p_k$ is the model threshold determined by $\Hh<sup>{p+1}\times\Sn<sup>{n-p-1}$. When $k=2$ and $n=m<sup>2$, we construct a smooth complete equality example on $\Sn<sup>n\setminus\Sn<sup>{(m<sup>2-m-2)/2}$. We also prove that the strict inequality $p&lt;p_k(n)$ holds under a finite positive linear-contact hypothesis.

• The paper proves that a smooth singular set of dimension p in a complete admissible conformal metric on S^n must satisfy p≤p_k(n), improving González’s earlier bound (n−2k)/2.
• The authors combine Jacobi-polynomial model algebra, cone separation, viscosity propagation, and a completeness contradiction to show that supercritical singular dimensions force finite-length escaping curves.
• For k=2, the paper constructs complete equality metrics when n=m², including S^9\S², while finite positive linear contact strengthens the conclusion to the strict bound p<p_k(n).

The paper by Li, Liu, and Ma (2608.16571) addresses a dimension obstruction for smooth singular sets of complete conformal metrics on the sphere with constant positive $\sigma_k$-curvature. For a closed smooth embedded $\Sigma^p\subset S^n$ with $1$g=v^{-2}g_{S^n}$ on $S^n\setminus\Sigma$ with $\lambda(g^{-1}A_g)\in\Gamma_k^+$ and $\sigma_k(g^{-1}A_g)\equiv\kappa>0$ must satisfy $p\le p_k(n)$, where $p_k(n)$ is the threshold dictated by the product model $H^{p+1}\times S^{n-p-1}$, not by the cruder bound $\Sigma^p\subset S^n$0 of González. The paper also constructs explicit equality examples for $\Sigma^p\subset S^n$1 when $\Sigma^p\subset S^n$2, and proves a strict inequality $\Sigma^p\subset S^n$3 under a finite positive linear-contact condition on the conformal factor.

Background and the model threshold

For $\Sigma^p\subset S^n$4 the problem reduces to the singular Yamabe (Loewner–Nirenberg) equation, whose singular-set theory goes back to Schoen–Yau's Hausdorff-dimension obstruction and the gluing constructions of Mazzeo–Pacard. For $\Sigma^p\subset S^n$5 the equation is fully nonlinear and elliptic only on the admissible branch $\Sigma^p\subset S^n$6; prior work includes radial classifications of Chang–Han–Yang, asymptotics of Han–Li–Teixeira, Delaunay-type ends of Mazzieri–Segatti, and viscosity-solution regularity of González–Li–Nguyen and Li–Nguyen–Xiong. The relevant prior dimension obstruction, due to González, states $\Sigma^p\subset S^n$7 under $\Sigma^p\subset S^n$8, $\Sigma^p\subset S^n$9.

The product model $1

$1

where $1symmetric functions of this eigenvalue vector, extended polynomially in $1$g=v^{-2}g_{S^n}$0 solutions below the threshold via gluing and conjectured $g=v^{-2}g_{S^n}$1 as the necessary condition for general $g=v^{-2}g_{S^n}$2; the paper establishes the non-strict version and shows the strict form is false in general.

The model algebra

A key reduction identifies $g=v^{-2}g_{S^n}$3 with values of a Jacobi-type polynomial family $g=v^{-2}g_{S^n}$4, $g=v^{-2}g_{S^n}$5, defined by a three-term recurrence with positive subdiagonal coefficients $g=v^{-2}g_{S^n}$6. Classical Jacobi-matrix theory gives real, simple, strictly interlacing zeros; if $g=v^{-2}g_{S^n}$7 is the largest zero of $g=v^{-2}g_{S^n}$8, then

$g=v^{-2}g_{S^n}$9

and positivity of $S^n\setminus\Sigma$0 for all $S^n\setminus\Sigma$1 is equivalent to $S^n\setminus\Sigma$2. The supercritical regime $S^n\setminus\Sigma$3 then yields a quantitative cone separation: writing $S^n\setminus\Sigma$4 for the perturbed eigenvalue vector of an upper test and $S^n\setminus\Sigma$5, there exist $S^n\setminus\Sigma$6 and $S^n\setminus\Sigma$7 such that $S^n\setminus\Sigma$8 for every $S^n\setminus\Sigma$9. The proof uses cone monotonicity $\lambda(g^{-1}A_g)\in\Gamma_k^+$0 to exclude $\lambda(g^{-1}A_g)\in\Gamma_k^+$1 from the closed cone on $\lambda(g^{-1}A_g)\in\Gamma_k^+$2, and the estimate $\lambda(g^{-1}A_g)\in\Gamma_k^+$3 for $\lambda(g^{-1}A_g)\in\Gamma_k^+$4. This uniform negative margin is the quantitative input that drives the entire dimension argument.

Mechanism of the dimension bound

The proof of the main theorem combines four ingredients.

Scale-invariant gradient estimate. Chang–Han–Yang ball convexity gives $\lambda(g^{-1}A_g)\in\Gamma_k^+$5, where $\lambda(g^{-1}A_g)\in\Gamma_k^+$6; consequently $\lambda(g^{-1}A_g)\in\Gamma_k^+$7 is uniformly Lipschitz in the logarithmic variable $\lambda(g^{-1}A_g)\in\Gamma_k^+$8.

Fermi-coordinate upper-contact analysis. For upper tests of the form $\lambda(g^{-1}A_g)\in\Gamma_k^+$9, the matrix $\sigma_k(g^{-1}A_g)\equiv\kappa>0$0 admits an exact radial-block decomposition in which $\sigma_k(g^{-1}A_g)\equiv\kappa>0$1, with $\sigma_k(g^{-1}A_g)\equiv\kappa>0$2 independent of $\sigma_k(g^{-1}A_g)\equiv\kappa>0$3. Membership $\sigma_k(g^{-1}A_g)\equiv\kappa>0$4 forces $\sigma_k(g^{-1}A_g)\equiv\kappa>0$5. A combinatorial identity expresses $\sigma_k(g^{-1}A_g)\equiv\kappa>0$6 through $\sigma_k(g^{-1}A_g)\equiv\kappa>0$7 and Newton-tensor contractions $\sigma_k(g^{-1}A_g)\equiv\kappa>0$8; the supercritical cone separation then implies that every admissible upper test satisfies

$\sigma_k(g^{-1}A_g)\equiv\kappa>0$9

for uniform constants $p\le p_k(n)$0.

One-dimensional propagation. A viscosity argument on the semiconvex peak functions $p\le p_k(n)$1 (maxima of $p\le p_k(n)$2 minus a quadratic tangential penalty over Fermi boxes) converts the test implication into a fixed increment $p\le p_k(n)$3 per logarithmic time interval of length $p\le p_k(n)$4: $p\le p_k(n)$5.

Completeness contradiction. The connecting curves between successive maximizers have $p\le p_k(n)$6-lengths bounded by $p\le p_k(n)$7, hence summable, while the base points converge to a point of $p\le p_k(n)$8 and $p\le p_k(n)$9. The concatenated curve has finite $p_k(n)$0-length, leaves every compact set of $p_k(n)$1, and has no limit point in the punctured sphere—contradicting completeness. Therefore $p_k(n)$2.

An important structural point is that the argument uses all the inequalities defining $p_k(n)$3, not only the equation $p_k(n)$4; the separation index $p_k(n)$5 selected by the model algebra need not equal $p_k(n)$6.

Strictness under finite linear contact

The equality examples (below) preclude a universal strict bound, but strictness is recovered under the hypothesis

$p_k(n)$7

The proof first shows this condition is gauge-invariant under stereographic projection. Blowing up at a sequence approaching the contact ratio yields a half-relaxed limit $p_k(n)$8 touched from above at a point by the linear test $p_k(n)$9, whose Schouten eigenvalues are $H^{p+1}\times S^{n-p-1}$0 with multiplicities $H^{p+1}\times S^{n-p-1}$1 and $H^{p+1}\times S^{n-p-1}$2. Cone monotonicity and continuity of $H^{p+1}\times S^{n-p-1}$3 along the contact sequence give $H^{p+1}\times S^{n-p-1}$4 for $H^{p+1}\times S^{n-p-1}$5. Since the main theorem gives $H^{p+1}\times S^{n-p-1}$6, equality would force $H^{p+1}\times S^{n-p-1}$7, a contradiction. Hence $H^{p+1}\times S^{n-p-1}$8.

The quadratic case and equality examples

For $H^{p+1}\times S^{n-p-1}$9 the algebra simplifies: $\Sigma^p\subset S^n$00, so $\Sigma^p\subset S^n$01, and the two model functions $\Sigma^p\subset S^n$02 are explicit polynomials in $\Sigma^p\subset S^n$03. The separation constant $\Sigma^p\subset S^n$04 and the implication $\Sigma^p\subset S^n$05 are computed directly.

The critical dimension $\Sigma^p\subset S^n$06 is an integer precisely when $\Sigma^p\subset S^n$07 ($\Sigma^p\subset S^n$08), giving $\Sigma^p\subset S^n$09. The paper constructs, for every $\Sigma^p\subset S^n$10, a smooth complete conformal metric on $\Sigma^p\subset S^n$11 with $\Sigma^p\subset S^n$12 and $\Sigma^p\subset S^n$13. The construction is a cohomogeneity-one ODE analysis on the model $\Sigma^p\subset S^n$14 (with $\Sigma^p\subset S^n$15, $\Sigma^p\subset S^n$16), for which $\Sigma^p\subset S^n$17 and $\Sigma^p\subset S^n$18. Writing $\Sigma^p\subset S^n$19 with $\Sigma^p\subset S^n$20, $\Sigma^p\subset S^n$21, the constant-$\Sigma^p\subset S^n$22 equation becomes a first-order system with an exact remainder $\Sigma^p\subset S^n$23 vanishing to first order at the origin. A Volterra fixed-point argument in a weighted Banach space produces the local solution branching off the collapsed orbit with $\Sigma^p\subset S^n$24; smoothness across $\Sigma^p\subset S^n$25 follows from a regular-singular lemma for equations of the form $\Sigma^p\subset S^n$26 under the spectral condition $\Sigma^p\subset S^n$27. Global existence is obtained by supersolution comparison, and the asymptotics give

$\Sigma^p\subset S^n$28

where $\Sigma^p\subset S^n$29. The conformal factor behaves as $\Sigma^p\subset S^n$30, producing a complete logarithmic end. The smallest instance is a complete metric on $\Sigma^p\subset S^n$31; further examples occur at $\Sigma^p\subset S^n$32.

A concluding remark asserts that an analogous cohomogeneity-one construction yields equality at every integral model endpoint $\Sigma^p\subset S^n$33 for general $\Sigma^p\subset S^n$34, so the bound $\Sigma^p\subset S^n$35 is optimal and cannot in general be strengthened to $\Sigma^p\subset S^n$36. This claim is stated without proof in the paper; the detailed construction is carried out only for $\Sigma^p\subset S^n$37.

Limitations and open questions

Three qualifications are explicit in the paper. First, the strict inequality requires the finite positive linear-contact hypothesis; whether strictness holds under weaker asymptotic control is not addressed. Second, the equality construction at general integral endpoints $\Sigma^p\subset S^n$38 is asserted by analogy with the $\Sigma^p\subset S^n$39 case rather than proved. Third, the theorem covers only smooth embedded singular sets and the range $\Sigma^p\subset S^n$40; the behavior for non-smooth singular sets, or for $\Sigma^p\subset S^n$41, and whether the threshold $\Sigma^p\subset S^n$42 is attained at non-integral values by non-product examples, remain open.

Conclusion

The paper establishes the model-threshold dimension bound $\Sigma^p\subset S^n$43 for smooth singular sets of complete, admissible, constant positive $\Sigma^p\subset S^n$44-curvature conformal metrics on the sphere, improving González's $\Sigma^p\subset S^n$45 bound to the sharp product-model threshold, and proves its optimality at integral endpoints via explicit $\Sigma^p\subset S^n$46 equality metrics on $\Sigma^p\subset S^n$47. The conditional strict bound under finite linear contact delineates precisely where the non-strict form is unavoidable.