Toric surfaces, K-stability and Calabi flow
Let X be a toric surface and u be a normalized symplectic potential on the corresponding polygon P . Suppose that the Riemannian curvature is bounded by a constant C 1 and ∫ ∂ P u d σ < C 2 , then there exists a constant C 3 depending only on C 1 , C 2 and P such that the diameter of X is bounded...
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| Veröffentlicht in: | Mathematische Zeitschrift 2014, Vol.276 (3-4), p.953-968 |
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| creator | Huang, Hongnian |
| description | Let
X
be a toric surface and
u
be a normalized symplectic potential on the corresponding polygon
P
. Suppose that the Riemannian curvature is bounded by a constant
C
1
and
∫
∂
P
u
d
σ
<
C
2
,
then there exists a constant
C
3
depending only on
C
1
,
C
2
and
P
such that the diameter of
X
is bounded by
C
3
. Moreoever, we can show that there is a constant
M
>
0
depending only on
C
1
,
C
2
and
P
such that Donaldson’s
M
-condition holds for
u
. As an application, we show that if
(
X
,
P
)
is (analytic) relative
K
-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow. |
| doi_str_mv | 10.1007/s00209-013-1228-8 |
| format | Article |
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X
be a toric surface and
u
be a normalized symplectic potential on the corresponding polygon
P
. Suppose that the Riemannian curvature is bounded by a constant
C
1
and
∫
∂
P
u
d
σ
<
C
2
,
then there exists a constant
C
3
depending only on
C
1
,
C
2
and
P
such that the diameter of
X
is bounded by
C
3
. Moreoever, we can show that there is a constant
M
>
0
depending only on
C
1
,
C
2
and
P
such that Donaldson’s
M
-condition holds for
u
. As an application, we show that if
(
X
,
P
)
is (analytic) relative
K
-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-013-1228-8</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Differential Geometry ; Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische Zeitschrift, 2014, Vol.276 (3-4), p.953-968</ispartof><rights>Springer-Verlag Berlin Heidelberg 2013</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-h1478-96923b30f61d6240cdba8a53d6d0e6950878ee171c2a6a8810c39d1f54d3f8773</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/s00209-013-1228-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00209-013-1228-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,776,780,881,27901,27902,41464,42533,51294</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00830729$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Huang, Hongnian</creatorcontrib><title>Toric surfaces, K-stability and Calabi flow</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>Let
X
be a toric surface and
u
be a normalized symplectic potential on the corresponding polygon
P
. Suppose that the Riemannian curvature is bounded by a constant
C
1
and
∫
∂
P
u
d
σ
<
C
2
,
then there exists a constant
C
3
depending only on
C
1
,
C
2
and
P
such that the diameter of
X
is bounded by
C
3
. Moreoever, we can show that there is a constant
M
>
0
depending only on
C
1
,
C
2
and
P
such that Donaldson’s
M
-condition holds for
u
. As an application, we show that if
(
X
,
P
)
is (analytic) relative
K
-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.</description><subject>Differential Geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><recordid>eNotkMFKxDAQhoMoWFcfwFuvotGZJG2mx6W4rljwsp5D2rRul7qVZFfZtzelnoZ__o9h-Bi7RXhEAP0UAAQUHFByFII4nbEElRQcSchzlsQ64xlpdcmuQtgBxFKrhN1vRt83aTj6zjZteEjfeDjYuh_6wym1e5eWdogx7Ybx95pddHYI7c3_XLCP1fOmXPPq_eW1XFZ8i0oTL_JCyFpCl6PLhYLG1ZZsJl3uoM2LDEhT26LGRtjcEiE0snDYZcrJjrSWC3Y3393awXz7_sv6kxltb9bLykw7AJKgRfGDkRUzGyK4_2y92Y1Hv4_vGQQzqTGzGhPVmEmNIfkH1fFToQ</recordid><startdate>2014</startdate><enddate>2014</enddate><creator>Huang, Hongnian</creator><general>Springer Berlin Heidelberg</general><general>Springer</general><scope>1XC</scope></search><sort><creationdate>2014</creationdate><title>Toric surfaces, K-stability and Calabi flow</title><author>Huang, Hongnian</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-h1478-96923b30f61d6240cdba8a53d6d0e6950878ee171c2a6a8810c39d1f54d3f8773</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Differential Geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Huang, Hongnian</creatorcontrib><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Huang, Hongnian</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Toric surfaces, K-stability and Calabi flow</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2014</date><risdate>2014</risdate><volume>276</volume><issue>3-4</issue><spage>953</spage><epage>968</epage><pages>953-968</pages><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>Let
X
be a toric surface and
u
be a normalized symplectic potential on the corresponding polygon
P
. Suppose that the Riemannian curvature is bounded by a constant
C
1
and
∫
∂
P
u
d
σ
<
C
2
,
then there exists a constant
C
3
depending only on
C
1
,
C
2
and
P
such that the diameter of
X
is bounded by
C
3
. Moreoever, we can show that there is a constant
M
>
0
depending only on
C
1
,
C
2
and
P
such that Donaldson’s
M
-condition holds for
u
. As an application, we show that if
(
X
,
P
)
is (analytic) relative
K
-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00209-013-1228-8</doi><tpages>16</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5874 |
| ispartof | Mathematische Zeitschrift, 2014, Vol.276 (3-4), p.953-968 |
| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_00830729v1 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Differential Geometry Mathematics Mathematics and Statistics |
| title | Toric surfaces, K-stability and Calabi flow |
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