MathRlife – Page 4 – Take notes of Mathematics

Van Abel May 23, 2023QuoteArXivChen-Yau, divergence, eigenvalue, inequality, Laplacian, Riemannian, Yang-type first inequality Leave a comment

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

Authors: Cristiano Silva, Juliana Miranda, Marcio Ara\’ujo Filho Categories: math.DG math.SP Comments: 15 pages \\ In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also … Continue reading “Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space”

Van Abel May 16, 2023ArXivconformal Gauss map, De Sitter space Leave a comment

Energy quantization for Willmore surfaces with bounded index

Title: Energy quantization for Willmore surfaces with bounded index Authors: Dorian Martino Categories: math.DG math.AP Comments: 49 pages \\ We prove an energy quantization result for Willmore surfaces with bounded index, whether the underlying Riemann surfaces degenerates in the moduli space or not. To do so, we translate the question on the conformal Gauss map’s point of view. In particular, we prove that in a neck or a collar region, the conformal Gauss map converges to a light-like geodesic in … Continue reading “Energy quantization for Willmore surfaces with bounded index”

Van Abel May 15, 2023ArXivblowup, 奇点, 曲线流, 渐近性, 空间曲线 Leave a comment

Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

Title: Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow Authors: Gabriel Khan Categories: math.DG math.AP Comments: 10 pages MSC-class: 53E10 53A04 \\ We study curve-shortening flow for twisted curves in $\mathbb{R}^3$ (i.e., curves with nowhere vanishing curvature $\kappa$ and torsion $\tau$) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type II … Continue reading “Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow”

Van Abel May 15, 2023ArXivBlaschke定理, 凸多面体, 曲率 Leave a comment

A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms

Title: A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms Authors: Alexander Borisenko and Vicente Miquel Categories: math.DG Comments: 11 pages, 2 figures MSC-class: 52A10, 52B99, 53C20 \\ Let $M$ be a $2$-space form. Let $P$ be a convex polygon in $M$. For these polygons, we define (and justify) a curvature $\kappa_i$ at each vertex $A_i$ of the polygon and and prove the following Blaschke’s type theorem: If $P$ is a convex plygon in $M$ with curvature at … Continue reading “A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms”

Van Abel May 15, 2023ArXivBrunn-Minkowski 理论, 复 Leave a comment

A Hilbert Bundles Description of Complex Brunn-Minkowski Theory

Title: A Hilbert Bundles Description of Complex Brunn-Minkowski Theory Authors: Tai Terje Huu Nguyen Categories: math.CV math.DG \\ The following is a Ph.D. thesis. The thesis is submitted in partial fulfillment of the requirements for the degree of Philosophiae Doctor (Ph.D.) at the Norwegian University of Science and Technology. \\ ( https://arxiv.org/abs/2305.07476 , 96kb) 复Brunn-Minkowski理论

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…”