For every integer $n\ge27$, we construct a smooth metric on $\mathbb S^n$ that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized $\sigma_2$-Yamabe equation admits a noncompact family of positive $\Gamma_2^+$-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the $\sigma_2$ problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension $n=27$.
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of t...
Using classical conformal divergence identities with a variable reference curvature, we prove four main results. First, we establish a general mean-curvature estimate for conformal metrics on the round hemisphere with $A_g\in\overline{\Gamma_2^+}$ and positive prescribed $H_2$ data. When $\sigma_2(A_g)=0$ and the bound...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We study the Yamabe--Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dim...
We establish, at every pole, the asymptotic expansion of the normalized $\Gamma_2$-Green function with a finite pole set in every dimension $n\ge5$. In dimensions $5\le n\le7$ the first correction is a unique nonnegative constant, whereas dimension $8$ exhibits a Weyl-driven $\sqrt{\log}$ term. For $n>8$ we construct t...
We construct a finite cyclic $L_3$ presentation (that is, a cyclic $L_\infty$ algebra with brackets vanishing above arity three) of the variational deformation theory defined by the heterotic $G_2$ superpotential near a torsion-free standard embedding, to first order in $\alpha'$. Auxiliary fields for the inverse cofra...
B. Brongers· 0 citations
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