ArXiv – MathRlife

Van Abel June 9, 2023ArXivAlmgren-Pitts, constant curvature, INTEREST, min-max Leave a comment

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)

Van Abel June 8, 2023ArXivAllard's regularity theorem, blowup, Brakke's regularity theorem, monotonicity formulas, MUSTREAD, varifolds, viscosity approach Leave a comment

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”

Van Abel June 8, 2023ArXivAlexandrov-Fenchel type inequalities, quantitative quermassintegral inequality, stability Leave a comment

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”

Van Abel June 8, 2023ArXivconvergence, Minkowski inequality, optimal constant Leave a comment

On the Minkowski inequality near the sphere

Title: On the Minkowski inequality near the sphere Authors: Otis Chodosh, Michael Eichmair, Thomas Koerber Categories: math.DG Comments: All comments welcome \\ We construct a sequence $\{\Sigma_\ell\}_{\ell=1}^\infty$ of closed, axially symmetric surfaces $\Sigma_\ell\subset \mathbb{R}^3$ that converges to the unit sphere in $W^{2,p}\cap C^1$ for every $p\in[1,\infty)$ and such that, for every $\ell$, $$ \int_{\Sigma_{\ell}}H_{\Sigma_\ell}-\sqrt{16\,\pi\,|\Sigma_{\ell}|}

Van Abel June 8, 2023ArXivbranch points, bubbles, conformally invariant Lagrangians, harmonic maps, Morse index, upper semi-continuity results, Willmore immersions Leave a comment

Morse Index Stability of Willmore Immersions I

Title: Morse Index Stability of Willmore Immersions I Authors: Alexis Michelat and Tristan Rivi\`ere Categories: math.DG math.AP Comments: 96 pages MSC-class: 35J35, 35J48, 35R01, 49Q10, 53A05, 53A10, 53A30, 53C42 \\ In a recent work, F. Da Lio, M. Gianocca, and T. Rivi\`ere developped a new method to show upper semi-continuity results in geometric analysis, which they applied to conformally invariant Lagrangians in dimension $2$ (that include harmonic maps). In this article, we apply this method to show that the sum … Continue reading “Morse Index Stability of Willmore Immersions I”

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…”