{"title":"非圆柱形三流形中奇异平面的存在性","authors":"Yongquan Zhang","doi":"10.4310/mrl.2023.v30.n2.a11","DOIUrl":null,"url":null,"abstract":"Let $P$ be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold $M$. Assume that $P^*=M^*\\cap P$ is nonempty, where $M^*$ is the interior of the convex core of $M$. Does this condition imply that $P$ is either closed or dense in $M$? A positive answer would furnish an analogue of Ratner's theorem in the infinite volume setting. In arXiv:1802.03853 it is shown that $P^*$ is either closed or dense in $M^*$. Moreover, there are at most countably many planes with $P^*$ closed, and in all previously known examples, $P$ was also closed in $M$. In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair $(M,P)$ such that $P^*$ is closed in $M^*$ but $P$ is not closed in $M$. In particular, the answer to the question above is no. Thus Ratner's theorem fails to generalize to planes in acylindrical 3-manifolds, without additional restrictions.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"6 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Existence of an exotic plane in an acylindrical 3-manifold\",\"authors\":\"Yongquan Zhang\",\"doi\":\"10.4310/mrl.2023.v30.n2.a11\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $P$ be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold $M$. Assume that $P^*=M^*\\\\cap P$ is nonempty, where $M^*$ is the interior of the convex core of $M$. Does this condition imply that $P$ is either closed or dense in $M$? A positive answer would furnish an analogue of Ratner's theorem in the infinite volume setting. In arXiv:1802.03853 it is shown that $P^*$ is either closed or dense in $M^*$. Moreover, there are at most countably many planes with $P^*$ closed, and in all previously known examples, $P$ was also closed in $M$. In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair $(M,P)$ such that $P^*$ is closed in $M^*$ but $P$ is not closed in $M$. In particular, the answer to the question above is no. Thus Ratner's theorem fails to generalize to planes in acylindrical 3-manifolds, without additional restrictions.\",\"PeriodicalId\":49857,\"journal\":{\"name\":\"Mathematical Research Letters\",\"volume\":\"6 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Research Letters\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4310/mrl.2023.v30.n2.a11\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Research Letters","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4310/mrl.2023.v30.n2.a11","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Existence of an exotic plane in an acylindrical 3-manifold
Let $P$ be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold $M$. Assume that $P^*=M^*\cap P$ is nonempty, where $M^*$ is the interior of the convex core of $M$. Does this condition imply that $P$ is either closed or dense in $M$? A positive answer would furnish an analogue of Ratner's theorem in the infinite volume setting. In arXiv:1802.03853 it is shown that $P^*$ is either closed or dense in $M^*$. Moreover, there are at most countably many planes with $P^*$ closed, and in all previously known examples, $P$ was also closed in $M$. In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair $(M,P)$ such that $P^*$ is closed in $M^*$ but $P$ is not closed in $M$. In particular, the answer to the question above is no. Thus Ratner's theorem fails to generalize to planes in acylindrical 3-manifolds, without additional restrictions.
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