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*, the nonnegative L2-normalized extremal exists and is unique when the first Dirichlet eigenvalue λD is sufficiently small. If the ambient volume is V and β=n(q−2)/2, the sharp constant satisfies CD=V1−q/2λD−β/2(1+O(λD)). For the normalized first eigenfunction φD, we obtain |uD−φD|H1=O(λD3/2) and CD/WMD(φD)=1+O(λD2), 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 L2-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.
Related
- Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
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- Sharp homogeneous Gagliardo--Nirenberg inequalities and normalized solutions for generalized stationary MMT equations
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BibTeX
@misc{mukherjee2026gagliardonirenberg,
title = {{Gagliardo--Nirenberg extremals on manifolds with small holes}},
author = {Mayukh Mukherjee and Utsab Sarkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32173},
eprint = {2610.32173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32173},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gagliardo--Nirenberg extremals on manifolds with small holes
AU - Mayukh Mukherjee
AU - Utsab Sarkar
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32173
UR - https://arxiv.org/abs/2610.32173
AB - 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^*, the nonnegative L^2-normalized extremal exists and is unique when the first Dirichlet eigenvalue λ_D is sufficiently small. If the ambient volume is V and β=n(q−2)/2, the sharp constant satisfies C_D=V^1−q/2λ_D^−β/2(1+O(λ_D)). For the normalized first eigenfunction φ_D, we obtain |u_D−φ_D|_H^1=O(λ_D^3/2) and C_D/W_M_D(φ_D)=1+O(λ_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.
ER -