Status – MathRlife

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 May 11, 2023StatusArXivcharacteristic classes, Pontryagin class, symplectic form Leave a comment

Title Classicfication Theory Authors Abel Milor Categories math…

Title: Classicfication Theory Authors: Abel Milor Categories: math.DG math.GR Comments: Master thesis. 69 pages, 8 figures \\ After introducing the simplicial manifolds, such as the different ways of defining the differential forms on them, we summarized a canonical way of calculating the characteristic classes of a $G$-principal bundle by computing them on the classifying bundle $EG\longrightarrow BG$. Finally, we calculated the first Pontryagin class on the classifying bundle of the Lie matrix groups and showed that for certain of them, … Continue reading “Title Classicfication Theory Authors Abel Milor Categories math…”

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 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…”