Quantitative stability of the total $Q$-curvature near minimizing metrics
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909248510754816 |
|---|---|
| author | Andrade, João Henrique König, Tobias Ratzkin, Jesse Wei, Juncheng |
| author_facet | Andrade, João Henrique König, Tobias Ratzkin, Jesse Wei, Juncheng |
| contents | Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \leq k < \frac{n}{2}$. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the $Q$-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case $k=1$. Next we prove, for any integer $1 \leq k< \frac{n}{2}$, the existence of an $n$-dimensional Riemannian manifold such that the $k$-th order $Q$-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06934 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative stability of the total $Q$-curvature near minimizing metrics Andrade, João Henrique König, Tobias Ratzkin, Jesse Wei, Juncheng Analysis of PDEs Differential Geometry Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \leq k < \frac{n}{2}$. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the $Q$-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case $k=1$. Next we prove, for any integer $1 \leq k< \frac{n}{2}$, the existence of an $n$-dimensional Riemannian manifold such that the $k$-th order $Q$-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field. |
| title | Quantitative stability of the total $Q$-curvature near minimizing metrics |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2407.06934 |