Automorphic Forms and Geometric Theories Robert Langlands Abstract. This is the translation of original article written in Russian, which discusses advanced mathematical concepts related to the Langlands program, automorphic forms, and differential geometry. https://publications.ias.edu/rpl/section/2659 Langlands_2022_On_the_analytic_form_of_the_geometric_theory_of_automorphic_forms

## Jensen不等式

回忆，定义在区间$I=(a,b)$上的函数$\varphi$称为凸函数, 如果对任意的$a < x < b$, $a < y < b$, 以及任意的$0\leq\lambda\leq1$, 成立如下不等式 \begin{equation} \varphi\left( (1-\lambda)x+\lambda y \right)\leq (1-\lambda)\varphi(x)+\lambda\varphi(y). \label{eq:convex} \end{equation} 从图形上， 假设$a < s < t < u < b$, 令 $t=(1-\lambda)s+\lambda u$, 则$\lambda= \frac{t-s}{u-s}$, $1-\lambda= \frac{u-t}{u-s}$, 从而\eqref{eq:convex}得到 \[ \varphi(t)\leq (1-\lambda)\varphi(s)+\lambda \varphi(u)\iff (1-\lambda)(\varphi(t)-\varphi(s))\leq \lambda \left( \varphi(u)-\varphi(t) \right), \] 故 \[ \frac{\varphi(t)-\varphi(s)}{t-s}\leq \frac{\varphi(u)-\varphi(t)}{u-t}. \]

## Existence of closed embedded curves of constant curvature via min-max

Title: Existence of closed embedded curves of constant curvature via min-max Authors: Lorenzo Sarnataro, Douglas Stryker Categories: math.DG math.DS Comments: 26 pages \\ We find conditions under which Almgren-Pitts min-max for the prescribed geodesic curvature functional in a closed oriented Riemannian surface produces a closed embedded curve of constant curvature. In particular, we find a closed embedded curve of any prescribed constant curvature in any metric on $S^2$ with $1/8$-pinched Gaussian curvature. \\ ( https://arxiv.org/abs/2306.04840 , 29kb)

## 边界爆破的脖子区域分解图

如下图所示，脖子区域的分解图为 \[ D_\delta^+\setminus D_{r_nR}^+(x_n) =D_\delta^+\setminus D_{\delta/2}^+(x_n’) \cup D_{\delta/2}^+(x_n’)\setminus D_{2d_n}^+(x_n’) \cup D_{2d_n}^+(x_n’)\setminus D_{d_n}^+(x_n) \cup D_{d_n}^+(x_n)\setminus D_{r_nR}^+(x_n). \]

## A short proof of Allard’s and Brakke’s regularity theorems

Title: A short proof of Allard’s and Brakke’s regularity theorems Authors: Guido De Philippis, Carlo Gasparetto, Felix Schulze Categories: math.AP math.DG \\ We give new short proofs of Allard’s regularity theorem for varifolds with bounded first variation and Brakke’s regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin. \\ … Continue reading “A short proof of Allard’s and Brakke’s regularity theorems”

## Stability of Alexandrov-Fenchel Type Inequalities for Nearly Spherical Sets in Space Forms

Title: Stability of Alexandrov-Fenchel Type Inequalities for Nearly Spherical Sets in Space Forms Authors: Rong Zhou and Tailong Zhou Categories: math.DG Comments: 25 pages MSC-class: 52A40, 53C42 \\ In this paper, we first derive a quantitative quermassintegral inequality for nearly spherical sets in $\mathbb{H}^{n+1}$ and $\mathbb{S}^{n+1}$, which is a generalization of the inequality proved in $\mathbb{R}^{n+1}$ [21]. Then we use this method to derive the stability of some geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets … Continue reading “Stability of Alexandrov-Fenchel Type Inequalities for Nearly Spherical Sets in Space Forms”