MathRlife – Page 5 – Take notes of Mathematics

Van Abel May 11, 2023StatusArXivABP, geometric inequalities Leave a comment

Title Some geometric inequalities by the ABP method…

Title: Some geometric inequalities by the ABP method Authors: Doanh Pham Categories: math.DG math.AP Comments: to appear in International Mathematics Research Notices \\ In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed $n$-dimensional minimal submanifolds $\Sigma$ of $\mathbb S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb S^n| \leq |\Sigma|$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for positive symmetric … Continue reading “Title Some geometric inequalities by the ABP method…”

Van Abel May 11, 2023杂记amsrefs, biber, Biblatex, LaTeX Leave a comment

为Biblatex添加类似amsrefs的样式表

首先克隆别人做好的样式表到一个用户目录，例如~/devlopement/tex/ git clone https://github.com/konn/biblatex-math.git 然后添加该目录到TexLive系统目录。首先查看系统目录： kpsewhich –var-value TEXMFHOME 如果该目录没有则创建, 并将你下载好的目录作软连接, 并查看是否连接好 mkdir ~/Library/texmf ln -s ~/devlopment/tex/biblatex-math/tex ~/Library/texmf/tex ls -la ~/Library/texmf/tex 最后刷新TexLive数据库 sudo mktexlsr 可以通过如下命令查看是否添加成功 kpsewhich math-alphabetic.bbx 如果能成功发现正确路径则表示添加成功。例如我的返回是： ~/Library/texmf/tex/latex/biblatex-math/bbx/math-alphabetic.bbx

Van Abel May 10, 2023StatusArXivblowup, conformal heat flow, evolution of metric, finite time singularity Leave a comment

Bubbling analysis of a conformal heat flow for…

Bubbling analysis of a conformal heat flow for harmonic maps Woongbae Park We study a conformal heat flow for harmonic maps. It is known that global weak solution of the flow exists and smooth except at mostly finitely many singular points. In this paper, we conduct a bubbling analysis for a finite time singularity.

Van Abel May 10, 2023StatusArXivAbelian Yang-Mills-Higgs, asymptotic behaviour, compactness, monotonicity formula, MUSTREAD, rectifiable varifold Leave a comment

The Yang Mills Higgs functional on complex line…

The Yang-Mills-Higgs functional on complex line bundles: asymptotics for critical points Giacomo Canevari, Federico Luigi Dipasquale, Giandomenico Orlandi We consider a gauge-invariant Ginzburg-Landau functional (also known as Abelian Yang-Mills-Higgs model) on Hermitian line bundles over closed Riemannian manifolds of dimension n≥3. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the non-self dual scaling, as the coupling parameter tends to zero. After a convenient choice of the gauge, we show compactness … Continue reading “The Yang Mills Higgs functional on complex line…”

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…”

Van Abel May 10, 2023StatusArXivchern classes, minimizing, Taubes construction, Yang-Mills Leave a comment

DIRECT MINIMIZING METHOD FOR YANG-MILLS ENERGY OVER SO(3) BUNDLE

DIRECT MINIMIZING METHOD FOR YANG-MILLS ENERGY OVER SO(3) BUNDLE In this paper, we use the direct minimizing method to find Yang- Mills connections for SO(3) bundles over closed four manifolds. By constructing test connections, we prove that a minimizing sequence converges strongly to a minimizer under certain assumptions. In case the strong convergence fails, we find an anti-selfdual (or selfdual) connection.