Related papers: Sharp quantitative stability of the M\"obius group…

In this note we present a short alternative proof of an estimate obtained by A.B.-Mantel, C.B. Muratov and T.M. Simon in [3] regarding the rigidity of degree {$\pm 1$} conformal maps of $\mathbb{S}^2$, i.e. its M\"obius transformations.

The purpose of this paper is to exhibit a quantitative stability result for the class of M\"obius transformations of $\mathbb{S}^{n-1}$ when $n\geq 3$. The main estimate is of local nature and asserts that for a Lipschitz map that is…

In any dimension $n\geq 3$, we prove an optimal stability estimate for the M\"obius group among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$, of the form $\inf_{\lambda>0,\phi\in \mathrm{M\"ob}(\mathbb S^{n-1})} \int_{\mathbb…

Liouville's theorem says that in dimension greater than two, all conformal maps are M\"obius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally…

We investigate both linear and nonlinear stability aspects of rigid motions (resp. M\"obius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from $\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for maps…

For degree $\pm 1$ harmonic maps from $\mathbb{R}^2$ (or $\mathbb{S}^2$) to $\mathbb{S}^2$, Bernand-Mantel, Muratov and Simon \cite{bernand2021quantitative} recently establish a uniformly quantitative stability estimate. Namely, for any map…

Among all metrics on $\mathbb S^d$ with $d>4$ that are conformal to the standard metric and have positive scalar curvature, the total $\sigma_2$-curvature, normalized by the volume, is uniquely (up to M\"obius transformations) minimized by…

Given a smooth closed Riemannian manifold $(M,g)$ of dimension $N \ge 3$, we derive sharp quantitative stability estimates for nonnegative functions near the solution set of the Yamabe problem on $(M,g)$. The seminal work of Struwe (1984)…

We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p<2$, while it…

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the…

We prove that if the minors of degree $k$ of a Sobolev map $\mathbb{R}^d \to \mathbb{R}^d$ are smooth then the map is smooth, when $k,d$ are not both even. We use this result to derive a simple, self-contained proof of the famous Liouville…

We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power…

A fundamental theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be constant. A remarkable Liouville-type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of $-\Delta u=u^{…

In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of M\"obius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension…

By developing a unified approach based on integral representations, we establish sharp quantitative stability estimates for critical points of the fractional Sobolev inequalities induced by the embedding $\dot{H}^s({\mathbb R}^n)…

In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate:…

Intermittent maps of Pomeau-Manneville type are well-studied in one-dimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that…

We consider the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree $1$ maps from closed…

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative…

In this paper, we prove a sharp quantitative stability result for the affine fractional \(L^2\)-Sobolev inequality in \(\dot H^s(\mathbb R^n)\), \(0<s<1\), introduced by Haddad--Ludwig (\emph{Math. Ann.} \textbf{388} (2024), 1091--1115). In…