Optimal geodesics for boundary points of the Gardiner–Masur compactification
The Gardiner–Masur compactification of Teichmüller space is naturally homeomorphic to the horofunction compactification of the Teichmüller metric, in the sense that the identity map on Teichmüller space extends to a homeomorphism. Let ξ and η be a pair of boundary points in the Gardiner–Masur compac...
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| Veröffentlicht in: | Geometriae dedicata 2024-12, Vol.218 (6), Article 110 |
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| creator | Lou, Xiaoke Su, Weixu Tan, Dong |
| description | The Gardiner–Masur compactification of Teichmüller space is naturally homeomorphic to the horofunction compactification of the Teichmüller metric, in the sense that the identity map on Teichmüller space extends to a homeomorphism. Let
ξ
and
η
be a pair of boundary points in the Gardiner–Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic that is optimal for the horofunctions corresponding to
ξ
and
η
. In particular, when
ξ
and
η
are Busemann points that fill up the surface, the Teichmüller geodesic converges to
ξ
in the forward direction and to
η
in the backward direction. As an application, we show that if
G
n
is a sequence of Teichmüller geodesics passing through
X
n
and
Y
n
such that
X
n
→
ξ
and
Y
n
→
η
in the Gardiner–Masur compactification, then
G
n
converges to a unique Teichmüller geodesic. |
| doi_str_mv | 10.1007/s10711-024-00952-w |
| format | Article |
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ξ
and
η
be a pair of boundary points in the Gardiner–Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic that is optimal for the horofunctions corresponding to
ξ
and
η
. In particular, when
ξ
and
η
are Busemann points that fill up the surface, the Teichmüller geodesic converges to
ξ
in the forward direction and to
η
in the backward direction. As an application, we show that if
G
n
is a sequence of Teichmüller geodesics passing through
X
n
and
Y
n
such that
X
n
→
ξ
and
Y
n
→
η
in the Gardiner–Masur compactification, then
G
n
converges to a unique Teichmüller geodesic.</description><identifier>ISSN: 0046-5755</identifier><identifier>EISSN: 1572-9168</identifier><identifier>DOI: 10.1007/s10711-024-00952-w</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Algebraic Geometry ; Convex and Discrete Geometry ; Differential Geometry ; Geodesy ; Hyperbolic Geometry ; Mathematics ; Mathematics and Statistics ; Original Paper ; Projective Geometry ; Topology</subject><ispartof>Geometriae dedicata, 2024-12, Vol.218 (6), Article 110</ispartof><rights>The Author(s), under exclusive licence to Springer Nature B.V. 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c200t-77e49e0e64c221a2e1a2cdad5ede8d8c093cb4239f91321a788788c8ed6a235b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10711-024-00952-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10711-024-00952-w$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Lou, Xiaoke</creatorcontrib><creatorcontrib>Su, Weixu</creatorcontrib><creatorcontrib>Tan, Dong</creatorcontrib><title>Optimal geodesics for boundary points of the Gardiner–Masur compactification</title><title>Geometriae dedicata</title><addtitle>Geom Dedicata</addtitle><description>The Gardiner–Masur compactification of Teichmüller space is naturally homeomorphic to the horofunction compactification of the Teichmüller metric, in the sense that the identity map on Teichmüller space extends to a homeomorphism. Let
ξ
and
η
be a pair of boundary points in the Gardiner–Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic that is optimal for the horofunctions corresponding to
ξ
and
η
. In particular, when
ξ
and
η
are Busemann points that fill up the surface, the Teichmüller geodesic converges to
ξ
in the forward direction and to
η
in the backward direction. As an application, we show that if
G
n
is a sequence of Teichmüller geodesics passing through
X
n
and
Y
n
such that
X
n
→
ξ
and
Y
n
→
η
in the Gardiner–Masur compactification, then
G
n
converges to a unique Teichmüller geodesic.</description><subject>Algebraic Geometry</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Geodesy</subject><subject>Hyperbolic Geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><subject>Projective Geometry</subject><subject>Topology</subject><issn>0046-5755</issn><issn>1572-9168</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kM1KAzEQx4MoWKsv4CngOZqPzWb3KMUvqPai55AmszWl3azJLqU338E39EmMruBNmGEuv_8M80PonNFLRqm6SowqxgjlBaG0lpzsDtCEScVJzcrqEE0oLUoilZTH6CSlNc2UUnyCnhZd77dmg1cQHCRvE25CxMswtM7EPe6Cb_uEQ4P7V8B3JjrfQvx8_3g0aYjYhm1nbO8bb03vQ3uKjhqzSXD2O6fo5fbmeXZP5ou7h9n1nFhOaU-UgqIGCmVhOWeGQ27rjJPgoHKVpbWwy4KLuqmZyICqqly2AlcaLuRSTNHFuLeL4W2A1Ot1GGKbT2rBWF1KISqWKT5SNoaUIjS6i_nZuNeM6m9vevSmszf9403vckiMoZThdgXxb_U_qS8f7HKE</recordid><startdate>20241201</startdate><enddate>20241201</enddate><creator>Lou, Xiaoke</creator><creator>Su, Weixu</creator><creator>Tan, Dong</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20241201</creationdate><title>Optimal geodesics for boundary points of the Gardiner–Masur compactification</title><author>Lou, Xiaoke ; Su, Weixu ; Tan, Dong</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-77e49e0e64c221a2e1a2cdad5ede8d8c093cb4239f91321a788788c8ed6a235b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Algebraic Geometry</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Geodesy</topic><topic>Hyperbolic Geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><topic>Projective Geometry</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Lou, Xiaoke</creatorcontrib><creatorcontrib>Su, Weixu</creatorcontrib><creatorcontrib>Tan, Dong</creatorcontrib><collection>CrossRef</collection><jtitle>Geometriae dedicata</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Lou, Xiaoke</au><au>Su, Weixu</au><au>Tan, Dong</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Optimal geodesics for boundary points of the Gardiner–Masur compactification</atitle><jtitle>Geometriae dedicata</jtitle><stitle>Geom Dedicata</stitle><date>2024-12-01</date><risdate>2024</risdate><volume>218</volume><issue>6</issue><artnum>110</artnum><issn>0046-5755</issn><eissn>1572-9168</eissn><abstract>The Gardiner–Masur compactification of Teichmüller space is naturally homeomorphic to the horofunction compactification of the Teichmüller metric, in the sense that the identity map on Teichmüller space extends to a homeomorphism. Let
ξ
and
η
be a pair of boundary points in the Gardiner–Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic that is optimal for the horofunctions corresponding to
ξ
and
η
. In particular, when
ξ
and
η
are Busemann points that fill up the surface, the Teichmüller geodesic converges to
ξ
in the forward direction and to
η
in the backward direction. As an application, we show that if
G
n
is a sequence of Teichmüller geodesics passing through
X
n
and
Y
n
such that
X
n
→
ξ
and
Y
n
→
η
in the Gardiner–Masur compactification, then
G
n
converges to a unique Teichmüller geodesic.</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s10711-024-00952-w</doi></addata></record> |
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| subjects | Algebraic Geometry Convex and Discrete Geometry Differential Geometry Geodesy Hyperbolic Geometry Mathematics Mathematics and Statistics Original Paper Projective Geometry Topology |
| title | Optimal geodesics for boundary points of the Gardiner–Masur compactification |
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