Curvature bound for $L_p$ Minkowski problem

[2304.11617] Curvature bound for $L_p$ Minkowski problem Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2304.11617 (math) [Submitted on 23 Apr 2023 ( v1 ), last revised 18 Sep 2024 (this version, v3)] Title: Curvature bound for $L_p$ Minkowski problem Authors: Kyeongsu Choi , Minhyun Kim , Taehun Lee View PDF HTML (experimental) Abstract: We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $\mu$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $\mathbb{R}^3$. Comments: 25 pages, to appear in Adv. Math Subjects: Differential Geometry (math.DG) ; Analysis of PDEs (math.AP) MSC classes: 53E99 (Primary) 35B65, 35C06, 35K96, 53A05 (Secondary) Cite as: arXiv:2304.11617 [math.DG] (or arXiv:2304.11617v3 [math.DG] for this version) https://doi.org/10.48550/arXiv.2304.11617 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Adv. Math. 458 (2024), part A, Paper No. 109959, 31 pp Related DOI : https://doi.org/10.1016/j.aim.2024.109959 Focus to learn more DOI(s) linking to related resources Submission history From: Taehun Lee [ view email ] [v1] Sun, 23 Apr 2023 11:21:18 UTC (25 KB) [v2] Wed, 3 May 2023 13:21:56 UTC (25 KB) [v3] Wed, 18 Sep 2024 00:25:23 UTC (25 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2023-04 Change to browse by: math math.AP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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