Equivariant Morse index of min–max G-invariant minimal hypersurfaces
For a closed Riemannian manifold M n + 1 with a compact Lie group G acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal G -invariant hypersurfaces provided 3 ≤ codim ( G · p ) ≤ 7 for all p ∈ M . In this paper, we show a compactness theorem...
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| Veröffentlicht in: | Mathematische annalen 2024-06, Vol.389 (2), p.1599-1637 |
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| creator | Wang, Tongrui |
| description | For a closed Riemannian manifold
M
n
+
1
with a compact Lie group
G
acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal
G
-invariant hypersurfaces provided
3
≤
codim
(
G
·
p
)
≤
7
for all
p
∈
M
. In this paper, we show a compactness theorem for these min–max minimal
G
-hypersurfaces and construct a
G
-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a
C
G
∞
-generic finiteness result for min–max
G
-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed
G
-invariant minimal hypersurface
Σ
⊂
M
constructed by the equivariant min–max on a
k
-dimensional homotopy class can be chosen to satisfy
Index
G
(
Σ
)
≤
k
. |
| doi_str_mv | 10.1007/s00208-023-02681-z |
| format | Article |
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M
n
+
1
with a compact Lie group
G
acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal
G
-invariant hypersurfaces provided
3
≤
codim
(
G
·
p
)
≤
7
for all
p
∈
M
. In this paper, we show a compactness theorem for these min–max minimal
G
-hypersurfaces and construct a
G
-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a
C
G
∞
-generic finiteness result for min–max
G
-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed
G
-invariant minimal hypersurface
Σ
⊂
M
constructed by the equivariant min–max on a
k
-dimensional homotopy class can be chosen to satisfy
Index
G
(
Σ
)
≤
k
.</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-023-02681-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Hyperspaces ; Invariants ; Lie groups ; Mathematics ; Mathematics and Statistics ; Riemann manifold ; Theorems</subject><ispartof>Mathematische annalen, 2024-06, Vol.389 (2), p.1599-1637</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. 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><citedby>FETCH-LOGICAL-c319t-eda9bfe446a5dfed298366a75378a06ecf13d0890e6223ddb714de0546e5652f3</citedby><cites>FETCH-LOGICAL-c319t-eda9bfe446a5dfed298366a75378a06ecf13d0890e6223ddb714de0546e5652f3</cites><orcidid>0000-0001-7608-2186</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/s00208-023-02681-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00208-023-02681-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Wang, Tongrui</creatorcontrib><title>Equivariant Morse index of min–max G-invariant minimal hypersurfaces</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>For a closed Riemannian manifold
M
n
+
1
with a compact Lie group
G
acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal
G
-invariant hypersurfaces provided
3
≤
codim
(
G
·
p
)
≤
7
for all
p
∈
M
. In this paper, we show a compactness theorem for these min–max minimal
G
-hypersurfaces and construct a
G
-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a
C
G
∞
-generic finiteness result for min–max
G
-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed
G
-invariant minimal hypersurface
Σ
⊂
M
constructed by the equivariant min–max on a
k
-dimensional homotopy class can be chosen to satisfy
Index
G
(
Σ
)
≤
k
.</description><subject>Hyperspaces</subject><subject>Invariants</subject><subject>Lie groups</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Riemann manifold</subject><subject>Theorems</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE1OwzAQhS0EEqVwAVaRWBvGv0mWqCoFqYgNrC03HkOqNmntBrVdcQduyEkwBMSOxWikme-9GT1CzhlcMoD8KgJwKChwkUoXjO4PyIBJwSkrID8kg7RXVBWCHZOTGOcAIADUgNyM1139akNtm01234aIWd043Gatz5Z18_H2vrTbbELr5hdK03ppF9nLboUhdsHbCuMpOfJ2EfHspw_J0834cXRLpw-Tu9H1lFaClRuKzpYzj1Jqq5xHx8tCaG1zJfLCgsbKM-GgKAE158K5Wc6kQ1BSo9KKezEkF73vKrTrDuPGzNsuNOmkEQmTSSZVonhPVaGNMaA3q5B-DjvDwHzlZfq8TMrLfOdl9kkkelFMcPOM4c_6H9UnsD1vQQ</recordid><startdate>20240601</startdate><enddate>20240601</enddate><creator>Wang, Tongrui</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-7608-2186</orcidid></search><sort><creationdate>20240601</creationdate><title>Equivariant Morse index of min–max G-invariant minimal hypersurfaces</title><author>Wang, Tongrui</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-eda9bfe446a5dfed298366a75378a06ecf13d0890e6223ddb714de0546e5652f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Hyperspaces</topic><topic>Invariants</topic><topic>Lie groups</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Riemann manifold</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Wang, Tongrui</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Wang, Tongrui</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Equivariant Morse index of min–max G-invariant minimal hypersurfaces</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2024-06-01</date><risdate>2024</risdate><volume>389</volume><issue>2</issue><spage>1599</spage><epage>1637</epage><pages>1599-1637</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>For a closed Riemannian manifold
M
n
+
1
with a compact Lie group
G
acting as isometries, the equivariant min–max theory gives the existence and the potential abundance of minimal
G
-invariant hypersurfaces provided
3
≤
codim
(
G
·
p
)
≤
7
for all
p
∈
M
. In this paper, we show a compactness theorem for these min–max minimal
G
-hypersurfaces and construct a
G
-invariant Jacobi field on the limit. Combining with an equivariant bumpy metrics theorem, we obtain a
C
G
∞
-generic finiteness result for min–max
G
-hypersurfaces with area uniformly bounded. As a main application, we further generalize the Morse index estimates for min–max minimal hypersurfaces to the equivariant setting. Namely, the closed
G
-invariant minimal hypersurface
Σ
⊂
M
constructed by the equivariant min–max on a
k
-dimensional homotopy class can be chosen to satisfy
Index
G
(
Σ
)
≤
k
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-023-02681-z</doi><tpages>39</tpages><orcidid>https://orcid.org/0000-0001-7608-2186</orcidid></addata></record> |
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| identifier | ISSN: 0025-5831 |
| ispartof | Mathematische annalen, 2024-06, Vol.389 (2), p.1599-1637 |
| issn | 0025-5831 1432-1807 |
| language | eng |
| recordid | cdi_proquest_journals_3054462245 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Hyperspaces Invariants Lie groups Mathematics Mathematics and Statistics Riemann manifold Theorems |
| title | Equivariant Morse index of min–max G-invariant minimal hypersurfaces |
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