harmonic maps – MathRlife

Van Abel June 8, 2023ArXivbranch points, bubbles, conformally invariant Lagrangians, harmonic maps, Morse index, upper semi-continuity results, Willmore immersions Leave a comment

Morse Index Stability of Willmore Immersions I

Title: Morse Index Stability of Willmore Immersions I Authors: Alexis Michelat and Tristan Rivi\`ere Categories: math.DG math.AP Comments: 96 pages MSC-class: 35J35, 35J48, 35R01, 49Q10, 53A05, 53A10, 53A30, 53C42 \\ In a recent work, F. Da Lio, M. Gianocca, and T. Rivi\`ere developped a new method to show upper semi-continuity results in geometric analysis, which they applied to conformally invariant Lagrangians in dimension $2$ (that include harmonic maps). In this article, we apply this method to show that the sum … Continue reading “Morse Index Stability of Willmore Immersions I”

Van Abel June 5, 2023MATHharmonic maps, Pohozaev Leave a comment

调和映射的Pohozaev恒等式

$\newcommand{\div}{\mathrm{div}\,}$ 我们知道调和映射的方程为 $$ \Delta u+A(u)(\nabla u,\nabla u)=0, $$ 其中$u:M^2\to N$是黎曼流形间的映射而$A(u)$是$N\hookrightarrow \mathbb{R}^n$在$u$处的第二基本形式。我们将用两种办法来证明如下的Pohozaev恒等式。 Theorem 1 (Pohozaev恒等式). 假设$u:M^2\to N$是光滑调和映射，则有 \[ \int_{\partial B_\rho} \lvert u_r \rvert^2=\int_{B_\rho}r^{-2} \lvert u_\theta \rvert^2. \]

Van Abel May 30, 2023ArXivharmonic maps, rigidity estimate Leave a comment

Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree

Title: Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree Authors: Melanie Rupflin Categories: math.AP math.DG MSC-class: 53C43, 58E20, 30C70, 26D10, \\ 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 necessarily close to a rational map. While such a rigidity statement turns out … Continue reading “Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree”

Van Abel May 10, 2023StatusArXivany time sequence, blowup, continuous in time, harmonic maps, MUSTREAD Leave a comment

Continuous in time bubble decomposition for the harmonic…

Continuous in time bubble decomposition for the harmonic map heat flow Jacek Jendrej, Andrew Lawrie, Wilhelm Schlag We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches … Continue reading “Continuous in time bubble decomposition for the harmonic…”