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SUMMARY:Martin de Borbon (Université de Nantes)
DTSTART;VALUE=DATE-TIME:20200709T120000Z
DTEND;VALUE=DATE-TIME:20200709T133000Z
DTSTAMP;VALUE=DATE-TIME:20230208T074403Z
UID:MathsSeminaratSH/1
DESCRIPTION:Title: Calabi-Yau metrics with cone singularities along intersecting com
plex lines: The unstable case\nby Martin de Borbon (Université de Nan
tes) as part of Maths Seminar at Shanghai\n\n\nAbstract\nAbstract: In coll
aboration with G. Edwards we produce (local) Calabi-Yau metrics\, in two c
omplex dimensions\, with cone singularities along intersecting complex lin
es\, for cone angles that strictly violate the Troyanov condition. We iden
tify the tangent cone at the origin as a product of two 2-cones. In the ta
ngent cone limit\, the line with the smallest cone angle remains apart whi
le the other lines collide into a single cone factor. \n\nTo prove our res
ult\, we first write an approximate solution with the desired asymptotic b
ehavior and small Ricci potential. The main difficulty is to invert the La
placian of such approximate solution metric in suitable Holder spaces. Onc
e this is done\, we use the implicit function theorem to perturb into an a
ctual Calabi-Yau metric.\n
LOCATION:https://researchseminars.org/talk/MathsSeminaratSH/1/
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SUMMARY:Jian Wang (Universität Augsburg)
DTSTART;VALUE=DATE-TIME:20200826T120000Z
DTEND;VALUE=DATE-TIME:20200826T140000Z
DTSTAMP;VALUE=DATE-TIME:20230208T074403Z
UID:MathsSeminaratSH/2
DESCRIPTION:Title: Topology of 3-manifolds with uniformly positive scalar curvature<
/a>\nby Jian Wang (Universität Augsburg) as part of Maths Seminar at Shan
ghai\n\n\nAbstract\nAbstract: One of fundamental questions is how to clas
sify open 3-manifolds with positive scalar curvature. The topology of open
3-manifolds is much complicated. For example\, Geometrization conjecture
is failed to be generalized to open 3-manifolds. In this talk\, we give a
classification for open 3-manifolds with uniformly positive scalar curvatu
re. Precisely\, we use minimal surface theory to give a prime decompositio
n for such manifolds.\n\nZoom Meeting\nPlease register in advance for thes
e meetings.\nAfter registration\, you will receive a confirmation email co
ntaining information about joining the meetings.\n
LOCATION:https://researchseminars.org/talk/MathsSeminaratSH/2/
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SUMMARY:Yu Li (Stony Brook University)
DTSTART;VALUE=DATE-TIME:20200827T130000Z
DTEND;VALUE=DATE-TIME:20200827T150000Z
DTSTAMP;VALUE=DATE-TIME:20230208T074403Z
UID:MathsSeminaratSH/3
DESCRIPTION:Title: Singularity models of the Ricci flow\nby Yu Li (Stony Brook U
niversity) as part of Maths Seminar at Shanghai\n\n\nAbstract\nAncient sol
utions model the singularity formation of the Ricci flow. In two and thre
e dimensions\, we currently have complete classifications for κ-noncollap
sed ancient solutions\, while the higher dimensional problem remains open.
This talk will survey recent developments of Ricci shrinkers\, which form
an important class of ancient solutions\, and higher dimensional ancient
solutions.\n
LOCATION:https://researchseminars.org/talk/MathsSeminaratSH/3/
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SUMMARY:Zakarias Sjöström Dyrefelt (ICTP Trieste)
DTSTART;VALUE=DATE-TIME:20200825T130000Z
DTEND;VALUE=DATE-TIME:20200825T150000Z
DTSTAMP;VALUE=DATE-TIME:20230208T074403Z
UID:MathsSeminaratSH/4
DESCRIPTION:Title: Optimal lower bounds for the J-functional and applications to exi
stence of cscK metrics\nby Zakarias Sjöström Dyrefelt (ICTP Trieste)
as part of Maths Seminar at Shanghai\n\n\nAbstract\nExistence of constant
scalar curvature Kähler (cscK) metrics on compact Kähler manifolds is a
central question in complex geometry. Following the variational approach
pioneered by Mabuchi in the 1980's it was recently proven (by X.X. Chen an
d J. Cheng) that existence of cscK metrics is equivalent to coercivity of
the Mabuchi K-energy functional on the space of Kähler metrics. In this t
alk I will present new coercivity estimates directly related to this probl
em\, focusing on the strongly related J-functional of Chen/Donaldson\, whi
ch occurs as the “energy part” in the Chen-Tian decomposition of the K
-energy\, and whose Euler-Lagrange equation is Donaldson’s J-equation.\n
\nAs a main result of the talk we give an explicit and optimal lower bound
for the J-functional\, in the sense of finding the largest possible const
ant in the definition of coercivity (which always exists and takes negativ
e values in general). This has applications to stability\, and sheds new l
ight on existence criteria for cscK metrics using Tian's alpha invariant\,
in the spirit of Dervan and Li-Shi-Yao. As a third application we explain
that there must always exist cscK metrics on compact Kähler manifolds wi
th nef canonical bundle\, thus on all smooth minimal models\, and also on
the blowup of any such manifold. This extends a result of Jian-Shi-Song wi
th a proof that does not depend on the Abundance conjecture.\n
LOCATION:https://researchseminars.org/talk/MathsSeminaratSH/4/
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