Gagliardo--Nirenberg extremals on manifolds with small holes

Abstract

We study the sharp homogeneous Dirichlet Gagliardo--Nirenberg quotient on a closed Riemannian manifold with holes of small capacity and connected complements. For each fixed exponent $2<q<2^{\ast }$, the nonnegative $L^2$-normalized extremal exists and is unique when the first Dirichlet eigenvalue ${\lambda }_D$ is sufficiently small. If the ambient volume is $V$ and $\beta =n(q-2)/2$, the sharp constant satisfies $C_D=V^{1-q/2}{\lambda }_D^{-\beta /2}(1+O({\lambda }_D))$. For the normalized first eigenfunction ${\phi }_D$, we obtain $\Vert u_D-{\phi }_D{\Vert }_{H^1}=O({\lambda }_D^{3/2})$ and $C_D/W_{M_D}({\phi }_D)=1+O({\lambda }_D^2)$, together with a quantitative stability estimate in terms of the relative deficit. The constants depend only on the ambient manifold and the exponent. On general compact manifolds with Dirichlet boundary conditions, we prove compactness above the Euclidean sharp constant and describe concentration at equality. Positive scalar curvature at one interior point implies attainment. For the $L^2$-critical nonlinear Schrödinger energy, the mass threshold for negative energy tends to zero as the capacity of the holes tends to zero, whereas the threshold for unboundedness below is unchanged.

arXiv comment17 pages, split off from arXiv:2601.22665v1. Comments highly appreciated!