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SUMMARY:Martin de Borbon (Université de Nantes)
DTSTART:20200709T120000Z
DTEND:20200709T133000Z
DTSTAMP:20260404T111101Z
UID:MathsSeminaratSH/1
DESCRIPTION:Title: Calabi-Yau metrics with cone singularities along intersect
ing complex lines: The unstable case\nby Martin de Borbon (Université
de Nantes) as part of Maths Seminar at Shanghai\n\n\nAbstract\nAbstract:
In collaboration with G. Edwards we produce (local) Calabi-Yau metrics\, i
n two complex dimensions\, with cone singularities along intersecting comp
lex lines\, for cone angles that strictly violate the Troyanov condition.
We identify the tangent cone at the origin as a product of two 2-cones. In
the tangent cone limit\, the line with the smallest cone angle remains ap
art while the other lines collide into a single cone factor. \n\nTo prove
our result\, we first write an approximate solution with the desired asymp
totic behavior and small Ricci potential. The main difficulty is to invert
the Laplacian of such approximate solution metric in suitable Holder spac
es. Once this is done\, we use the implicit function theorem to perturb in
to an actual Calabi-Yau metric.\n
LOCATION:https://stable.researchseminars.org/talk/MathsSeminaratSH/1/
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SUMMARY:Jian Wang (Universität Augsburg)
DTSTART:20200826T120000Z
DTEND:20200826T140000Z
DTSTAMP:20260404T111101Z
UID:MathsSeminaratSH/2
DESCRIPTION:Title: Topology of 3-manifolds with uniformly positive scalar cur
vature\nby Jian Wang (Universität Augsburg) as part of Maths Seminar
at Shanghai\n\n\nAbstract\nAbstract: One of fundamental questions is how
to classify open 3-manifolds with positive scalar curvature. The topology
of open 3-manifolds is much complicated. For example\, Geometrization conj
ecture is failed to be generalized to open 3-manifolds. In this talk\, we
give a classification for open 3-manifolds with uniformly positive scalar
curvature. Precisely\, we use minimal surface theory to give a prime decom
position for such manifolds.\n\nZoom Meeting\nPlease register in advance f
or these meetings.\nAfter registration\, you will receive a confirmation e
mail containing information about joining the meetings.\n
LOCATION:https://stable.researchseminars.org/talk/MathsSeminaratSH/2/
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SUMMARY:Yu Li (Stony Brook University)
DTSTART:20200827T130000Z
DTEND:20200827T150000Z
DTSTAMP:20260404T111101Z
UID:MathsSeminaratSH/3
DESCRIPTION:Title: Singularity models of the Ricci flow\nby Yu Li (Stony
Brook University) as part of Maths Seminar at Shanghai\n\n\nAbstract\nAnci
ent solutions model the singularity formation of the Ricci flow. In two a
nd three dimensions\, we currently have complete classifications for κ-no
ncollapsed ancient solutions\, while the higher dimensional problem remain
s open. This talk will survey recent developments of Ricci shrinkers\, whi
ch form an important class of ancient solutions\, and higher dimensional a
ncient solutions.\n
LOCATION:https://stable.researchseminars.org/talk/MathsSeminaratSH/3/
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SUMMARY:Zakarias Sjöström Dyrefelt (ICTP Trieste)
DTSTART:20200825T130000Z
DTEND:20200825T150000Z
DTSTAMP:20260404T111101Z
UID:MathsSeminaratSH/4
DESCRIPTION:Title: Optimal lower bounds for the J-functional and applications
to existence of cscK metrics\nby Zakarias Sjöström Dyrefelt (ICTP T
rieste) as part of Maths Seminar at Shanghai\n\n\nAbstract\nExistence of c
onstant scalar curvature Kähler (cscK) metrics on compact Kähler manifol
ds is a central question in complex geometry. Following the variational ap
proach pioneered by Mabuchi in the 1980's it was recently proven (by X.X.
Chen and J. Cheng) that existence of cscK metrics is equivalent to coerciv
ity of the Mabuchi K-energy functional on the space of Kähler metrics. In
this talk I will present new coercivity estimates directly related to thi
s problem\, focusing on the strongly related J-functional of Chen/Donaldso
n\, which occurs as the “energy part” in the Chen-Tian decomposition o
f the K-energy\, and whose Euler-Lagrange equation is Donaldson’s J-equa
tion.\n\nAs a main result of the talk we give an explicit and optimal lowe
r bound for the J-functional\, in the sense of finding the largest possibl
e constant in the definition of coercivity (which always exists and takes
negative values in general). This has applications to stability\, and shed
s new light on existence criteria for cscK metrics using Tian's alpha inva
riant\, 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 manif
olds with nef canonical bundle\, thus on all smooth minimal models\, and a
lso on the blowup of any such manifold. This extends a result of Jian-Shi-
Song with a proof that does not depend on the Abundance conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/MathsSeminaratSH/4/
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