On the fill-in of nonnegative scalar curvature metrics
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data ( Σ , γ , H ) . We prove that given a metric γ on S n - 1 ( 3 ≤ n ≤ 7 ), ( S n - 1 , γ , H ) admits no fill-in of NNSC metrics provided the prescribed mean...
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| Veröffentlicht in: | Mathematische annalen 2021-02, Vol.379 (1-2), p.235-270 |
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| creator | Shi, Yuguang Wang, Wenlong Wei, Guodong Zhu, Jintian |
| description | In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data
(
Σ
,
γ
,
H
)
. We prove that given a metric
γ
on
S
n
-
1
(
3
≤
n
≤
7
),
(
S
n
-
1
,
γ
,
H
)
admits no fill-in of NNSC metrics provided the prescribed mean curvature
H
is large enough (Theorem 4). Moreover, we prove that if
γ
is a positive scalar curvature (PSC) metric isotopic to the standard metric on
S
n
-
1
, then the much weaker condition that the total mean curvature
∫
S
n
-
1
H
d
μ
γ
is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the
θ
-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins. |
| doi_str_mv | 10.1007/s00208-020-02087-1 |
| format | Article |
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(
Σ
,
γ
,
H
)
. We prove that given a metric
γ
on
S
n
-
1
(
3
≤
n
≤
7
),
(
S
n
-
1
,
γ
,
H
)
admits no fill-in of NNSC metrics provided the prescribed mean curvature
H
is large enough (Theorem 4). Moreover, we prove that if
γ
is a positive scalar curvature (PSC) metric isotopic to the standard metric on
S
n
-
1
, then the much weaker condition that the total mean curvature
∫
S
n
-
1
H
d
μ
γ
is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the
θ
-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-020-02087-1</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Curvature ; Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische annalen, 2021-02, Vol.379 (1-2), p.235-270</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2020</rights><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-d30e748c66f9ba748a83d0a2dcc7e5b436488dbc8fc18838d82b268248c8001f3</citedby><cites>FETCH-LOGICAL-c319t-d30e748c66f9ba748a83d0a2dcc7e5b436488dbc8fc18838d82b268248c8001f3</cites><orcidid>0000-0003-3088-046X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00208-020-02087-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00208-020-02087-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Shi, Yuguang</creatorcontrib><creatorcontrib>Wang, Wenlong</creatorcontrib><creatorcontrib>Wei, Guodong</creatorcontrib><creatorcontrib>Zhu, Jintian</creatorcontrib><title>On the fill-in of nonnegative scalar curvature metrics</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data
(
Σ
,
γ
,
H
)
. We prove that given a metric
γ
on
S
n
-
1
(
3
≤
n
≤
7
),
(
S
n
-
1
,
γ
,
H
)
admits no fill-in of NNSC metrics provided the prescribed mean curvature
H
is large enough (Theorem 4). Moreover, we prove that if
γ
is a positive scalar curvature (PSC) metric isotopic to the standard metric on
S
n
-
1
, then the much weaker condition that the total mean curvature
∫
S
n
-
1
H
d
μ
γ
is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the
θ
-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.</description><subject>Curvature</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LAzEQxYMoWKtfwFPAc3SSbHanRylqhUIveg7ZbFK3bLM12S347U1dwZuH-QPzfm_gEXLL4Z4DVA8JQACy3E6FFeNnZMYLKRhHqM7JLN8VUyj5JblKaQcAEkDNSLkJdPhw1Lddx9pAe09DH4LbmqE9Opqs6UykdoxHM4zR0b0bYmvTNbnwpkvu5nfOyfvz09tyxdabl9fl45pZyRcDayS4qkBbln5Rm7wZlA0Y0VhbOVUXsiwQm9qitxxRYoOiFiWKjCAA93JO7ibfQ-w_R5cGvevHGPJLnUWlAiVFkVViUtnYpxSd14fY7k380hz0KR895aNz0z_5aJ4hOUEpi8PWxT_rf6hvY8FmvA</recordid><startdate>20210201</startdate><enddate>20210201</enddate><creator>Shi, Yuguang</creator><creator>Wang, Wenlong</creator><creator>Wei, Guodong</creator><creator>Zhu, Jintian</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0003-3088-046X</orcidid></search><sort><creationdate>20210201</creationdate><title>On the fill-in of nonnegative scalar curvature metrics</title><author>Shi, Yuguang ; Wang, Wenlong ; Wei, Guodong ; Zhu, Jintian</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-d30e748c66f9ba748a83d0a2dcc7e5b436488dbc8fc18838d82b268248c8001f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Curvature</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shi, Yuguang</creatorcontrib><creatorcontrib>Wang, Wenlong</creatorcontrib><creatorcontrib>Wei, Guodong</creatorcontrib><creatorcontrib>Zhu, Jintian</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shi, Yuguang</au><au>Wang, Wenlong</au><au>Wei, Guodong</au><au>Zhu, Jintian</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the fill-in of nonnegative scalar curvature metrics</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2021-02-01</date><risdate>2021</risdate><volume>379</volume><issue>1-2</issue><spage>235</spage><epage>270</epage><pages>235-270</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data
(
Σ
,
γ
,
H
)
. We prove that given a metric
γ
on
S
n
-
1
(
3
≤
n
≤
7
),
(
S
n
-
1
,
γ
,
H
)
admits no fill-in of NNSC metrics provided the prescribed mean curvature
H
is large enough (Theorem 4). Moreover, we prove that if
γ
is a positive scalar curvature (PSC) metric isotopic to the standard metric on
S
n
-
1
, then the much weaker condition that the total mean curvature
∫
S
n
-
1
H
d
μ
γ
is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the
θ
-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-020-02087-1</doi><tpages>36</tpages><orcidid>https://orcid.org/0000-0003-3088-046X</orcidid></addata></record> |
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| issn | 0025-5831 1432-1807 |
| language | eng |
| recordid | cdi_proquest_journals_2486505324 |
| source | Springer Nature - Complete Springer Journals |
| subjects | Curvature Mathematics Mathematics and Statistics |
| title | On the fill-in of nonnegative scalar curvature metrics |
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