{"title":"Nonpositively Curved Surfaces are Loewner","authors":"Mikhail G. Katz, Stéphane Sabourau","doi":"10.1007/s12220-024-01732-4","DOIUrl":null,"url":null,"abstract":"<p>We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area.\n</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01732-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area.
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