Sharp thresholds for the Escobar functional: the Escobar-Willmore mass, geometric selection, and compactness trichotomy
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866916019325370368 |
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| author | Mukherjee, Mayukh Sarkar, Utsab |
| author_facet | Mukherjee, Mayukh Sarkar, Utsab |
| contents | We study the hemisphere threshold for the conformally covariant Escobar functional on compact Riemannian manifolds $(M^n,g)$ with boundary. The near-threshold landscape is organized by boundary invariants: the first-order coefficient $ρ_n^{\mathrm{conf}}H_g$ vanishes identically, so the leading obstruction is a renormalized boundary mass $\mathfrak R_g$ (second order, $n\ge5$), followed by a cubic invariant $Θ_g$ (third order, $n\ge6$), with a Green kernel interaction $\mathsf G_\partial$ in the multi-bubble regime.
Exact evaluation of weighted profile moments yields $κ_1=κ_2=0$: the coefficients of $\operatorname{Ric}_g(ν,ν)$ and $\mathrm{Scal}_{\bar g}$ in the bare mass vanish. On $\{H_g=0\}$ the mass reduces to $\mathfrak R_g^{\mathrm{bare}}=\frac{6-n}{2(n-1)(n-3)(n-4)}|\mathring{\mathrm{II}}|^2$. The Lyapunov--Schmidt correction gives $\mathfrak R_g^{\mathrm{red}}\le\mathfrak R_g^{\mathrm{bare}}\le0$ for $n\ge6$; for $n=5$ the nonlocal back-reaction overcomes the positive bare coefficient. In every dimension $n\ge5$, non-umbilic boundaries are automatically subcritical: $C^*_{\mathrm{Esc}}<S_\ast$ whenever $\mathring{\mathrm{II}}\neq0$.
At threshold, on manifolds not conformally diffeomorphic to the hemisphere, every blow-up of positive constrained critical points is one-bubble and concentrates at an umbilic point with $\mathfrak R_g=0$, $\nabla_\partial\mathfrak R_g=0$. Since $\mathfrak R_g^{\mathrm{red}}<0$ at every non-umbilic point, threshold concentration occurs only on the umbilic stratum $\{\mathring{\mathrm{II}}=0\}$. There $Θ_g$ governs the next bifurcation: $Θ_g<0$ forces subcriticality; for $n\ge7$, $Θ_g>0$ yields compactness and hemispherical rigidity. In the multi-bubble regime we establish global compactness at Escobar multiples with equal-mass quantization and conditional exclusion of pure multi-bubbling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_22665 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp thresholds for the Escobar functional: the Escobar-Willmore mass, geometric selection, and compactness trichotomy Mukherjee, Mayukh Sarkar, Utsab Differential Geometry Analysis of PDEs Classical Analysis and ODEs 53C21, 35B44, 35J25 We study the hemisphere threshold for the conformally covariant Escobar functional on compact Riemannian manifolds $(M^n,g)$ with boundary. The near-threshold landscape is organized by boundary invariants: the first-order coefficient $ρ_n^{\mathrm{conf}}H_g$ vanishes identically, so the leading obstruction is a renormalized boundary mass $\mathfrak R_g$ (second order, $n\ge5$), followed by a cubic invariant $Θ_g$ (third order, $n\ge6$), with a Green kernel interaction $\mathsf G_\partial$ in the multi-bubble regime. Exact evaluation of weighted profile moments yields $κ_1=κ_2=0$: the coefficients of $\operatorname{Ric}_g(ν,ν)$ and $\mathrm{Scal}_{\bar g}$ in the bare mass vanish. On $\{H_g=0\}$ the mass reduces to $\mathfrak R_g^{\mathrm{bare}}=\frac{6-n}{2(n-1)(n-3)(n-4)}|\mathring{\mathrm{II}}|^2$. The Lyapunov--Schmidt correction gives $\mathfrak R_g^{\mathrm{red}}\le\mathfrak R_g^{\mathrm{bare}}\le0$ for $n\ge6$; for $n=5$ the nonlocal back-reaction overcomes the positive bare coefficient. In every dimension $n\ge5$, non-umbilic boundaries are automatically subcritical: $C^*_{\mathrm{Esc}}<S_\ast$ whenever $\mathring{\mathrm{II}}\neq0$. At threshold, on manifolds not conformally diffeomorphic to the hemisphere, every blow-up of positive constrained critical points is one-bubble and concentrates at an umbilic point with $\mathfrak R_g=0$, $\nabla_\partial\mathfrak R_g=0$. Since $\mathfrak R_g^{\mathrm{red}}<0$ at every non-umbilic point, threshold concentration occurs only on the umbilic stratum $\{\mathring{\mathrm{II}}=0\}$. There $Θ_g$ governs the next bifurcation: $Θ_g<0$ forces subcriticality; for $n\ge7$, $Θ_g>0$ yields compactness and hemispherical rigidity. In the multi-bubble regime we establish global compactness at Escobar multiples with equal-mass quantization and conditional exclusion of pure multi-bubbling. |
| title | Sharp thresholds for the Escobar functional: the Escobar-Willmore mass, geometric selection, and compactness trichotomy |
| topic | Differential Geometry Analysis of PDEs Classical Analysis and ODEs 53C21, 35B44, 35J25 |
| url | https://arxiv.org/abs/2601.22665 |