Related papers: Quantitative stability of harmonic maps from $\mat…

We consider half-harmonic maps from $\mathbb{R}$ (or $\mathbb{S}$) to $\mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A…

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.

We show some preservation results of amenably extending strongly Ulam stable groups under mild decay assumptions, including quantitative preservation of asymptotic bounds under the assumption that the modulus of stability is H\"older…

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…

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target…

We show how a rigidity estimate introduced in recent work of Bernand-Mantel, Muratov and Simon can be derived using the harmonic map flow.

In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider H^1_loc-maps u defined on a parabolic ball P\subset M\times R and with target manifold N, that have bounded Dirichlet-energy…

In this work we prove a sharp quantitative form of Liouville's theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$ with degree $\pm 1$ are M\"obius transformations. In the case $n=3$ this estimate…

Let $\Omega\subset \mathbb{R}^d$ be an open set of finite measure and let $\Theta$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $\Omega$ when its second eigenvalue is close to…

In this paper, we will show the Yau's gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $\kappa$, $\kappa\geq0$, in the sense of Alexandrov. As a direct application, it gives some…

In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a…

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy…

In this paper, we prove that the large energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation.

A group $\Gamma$ is said to be uniformly HS stable if any map $\varphi : \Gamma \to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We…

As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4\pi \vert deg(v)\vert$ with equality if and only if $v$ is a rational map one might ask whether maps with small energy defect $\delta_v=E(v)-4\pi \vert deg(v)\vert$ are…

This paper deals with quantitative spectral stability for compact operators acting on $L^2(X,m)$, where $(X,m)$ is a measure space. Under fairly general assumptions, we provide a characterization of the dominant term of the asymptotic…

For an harmonic map $u$ from a domain $U\subset{\rm X}$ in an ${\sf RCD}(K,N)$ space ${\rm X}$ to a ${\sf CAT}(0)$ space ${\rm Y}$ we prove the Lipschitz estimate \[ {\rm Lip}(u|_B)\leq \frac {C(K^-R^2,N)}r\inf_{{\sf o}\in {\rm…

We show a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic…

In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and H\"ormander. We provide local stability estimates that…

We establish sharp quantitative stability estimates near finite sums of ground states. The results depend on the dimension and the order of nonlinearity.