MathRlife – Page 3 – Take notes of Mathematics

Van Abel June 1, 2023StatusArXivPenrose Inequality, sharp mass-capacity inequality Leave a comment

Title The Sharp $p$ Penrose Inequality Authors Liam…

Title: The Sharp $p$-Penrose Inequality Authors: Liam Mazurowski, Xuan Yao Categories: math.DG math-ph math.CA math.MP Comments: 19 pages, comments are welcome! \\ Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $\Sigma$. Fix a number $1 < p < 2$. We prove a sharp mass-capacity inequality relating the ADM mass of $M$ with the $p$-capacity of $\Sigma$ in $M$. Equality holds if and only if $M$ is isometric to a spatial Schwarzschild manifold with … Continue reading “Title The Sharp $p$ Penrose Inequality Authors Liam…”

Van Abel June 1, 2023StatusArXivbubble, Heintze-Karcher inequality, Pohozaev identity, Serrin's type problem Leave a comment

Title A note on Serrin’s type problem on…

Title: A note on Serrin’s type problem on Riemannian manifolds Authors: Allan Freitas, Alberto Roncoroni and M\’arcio Santos Categories: math.DG math.AP Comments: Comments are welcome! \\ In this paper, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach a Serrin problem in bounded domains of manifolds endowed with a closed conformal vector field. … Continue reading “Title A note on Serrin’s type problem on…”

Van Abel June 1, 2023ArXiv$\Phi_{(3)}$-conservation law, $\phi_3$-harmonic maps, extrinsic average variational method, first variation formula, Liouville type, monotonicity formula, MUSTREAD, second variation formula, sphere theorems, stress-energy tensor, topological vanishing theorems, Unstable manifolds Leave a comment

The geometry of $\Phi_{(3)}$-harmonic maps

Title: The geometry of $\Phi_{(3)}$-harmonic maps Authors: Shuxiang Feng, Yingbo Han, Kaige Jiang and Shihshu Walter Wei Categories: math.DG math-ph math.AP math.MP Comments: 46 pages, to appear in Nonlinear Analysis (2023). arXiv admin note: text overlap with arXiv:1911.05855 MSC-class: 58E20, 53C21, 53C25 \\ In this paper, we motivate and extend the study of harmonic maps or $\Phi_{(1)}$-harmonic maps (cf [15], Remark 1.3 (iii)), $\Phi$-harmonic maps or $\Phi_{(2)}$-harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric properties of $\Phi_{(3)}$-harmonic maps … Continue reading “The geometry of $\Phi_{(3)}$-harmonic maps”

Van Abel May 30, 2023杂记Octave Leave a comment

在线计算工具

1. 包的安装与使用 Octave是一个类似MatLab的在线计算工具。在Linux/Mac下还有命令行工具。这里我用Mac举例如下： octave pkg install -forge symbolic pkg load symbolic syms t p syms expr = cos(t)*cos(p)+sin(t)*sin(p) simplify(expr) expr = cos(t)sin(p)-sin(t)cos(p) simplify(expr)

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 30, 2023ArXivHiggs, Hitchin, hyperkähler Leave a comment

Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles

Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles Takuro Mochizuki The moduli space of stable Higgs bundles of degree 0 is equipped with the hyperkähler metric, called the Hitchin metric. On the locus where the Hitchin fibration is smooth, there is the hyperkähler metric called the semi-flat metric, associated with the algebraic integrable systems with the Hitchin section. We prove the exponentially rapid decay of the difference between the Hitchin metric and the semi-flat metric … Continue reading “Asymptotic behaviour of the Hitchin metric on the moduli space of Higgs bundles”