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UID:eb274aba3f0b2cf27ea54aff5b296c4a
CATEGORIES:Nonlinear Analysis
CREATED:20260211T165949
SUMMARY:Yannick Sire: Harmonic maps into singular spaces
DESCRIPTION:Abstract: The heat flow of harmonic maps from a smooth, compact Riemannian 
 manifold without boundary, (M,g) into another smooth, compact Riemannian ma
 nifold without boundary (N,h) was first studied in the seminal work of Eell
 s and Sampson when the target manifold (N,h) has non-positive curvature. R.
 Hamilton studied the case when M has a compact  smooth boundary (and some s
 pecial cases when N has also a smooth, compact boundary).  Gromov-Schoen st
 udied harmonic maps from M  into a singular Cat(0) space which was used to 
  understand the p-adic superrigidity of lattices in groups of rank one.  A 
 key analytical property of such harmonic maps is the Lipschitz continuity, 
 from which one derives  Bochner type estimates and vanishing theorems.  As 
 for Eells-Sampson theorem, it is rather natural to study the associated gra
 dient (heat) flow,  and it has been a long open problem to construct suitab
 le weak solutions in the singular setting.  In this talk, I shall describe 
 an elliptic approach (which goes back to De Giorgi and also T. Ilmanen in t
 he 1990s) to this problem both in the smooth and the singular settings, i.e
 . when the target is CAT(0) space. I will explain how to get Lipschitz boun
 ds in the space variables (hence a suitable solution of the flow) and how t
 his new approach offers as well a new viewpoint on the old problem of mappi
 ngs between smooth manifolds. This is joint work with FH Lin, A Segatti and
  C Wang.\n
X-ALT-DESC;FMTTYPE=text/html:<p>Abstract: The heat flow of harmonic maps from a smooth, compact Riemanni
 an manifold without boundary, (M,g) into another smooth, compact Riemannian
  manifold without boundary (N,h) was first studied in the seminal work of E
 ells and Sampson when the target manifold (N,h) has non-positive curvature.
  R.Hamilton studied the case when M has a compact&nbsp; smooth boundary (an
 d some special cases when N has also a smooth, compact boundary).&nbsp; Gro
 mov-Schoen studied harmonic maps from M&nbsp; into a singular Cat(0) space 
 which was used to&nbsp; understand the p-adic superrigidity of lattices in 
 groups of rank one.&nbsp; A key analytical property of such harmonic maps i
 s the Lipschitz continuity, from which one derives&nbsp; Bochner type estim
 ates and vanishing theorems.&nbsp; As for Eells-Sampson theorem, it is rath
 er natural to study the associated gradient (heat) flow,&nbsp; and it has b
 een a long open problem to construct suitable weak solutions in the singula
 r setting.&nbsp; In this talk, I shall describe an elliptic approach (which
  goes back to De Giorgi and also T. Ilmanen in the 1990s) to this problem b
 oth in the smooth and the singular settings, i.e. when the target is CAT(0)
  space. I will explain how to get Lipschitz bounds in the space variables (
 hence a suitable solution of the flow) and how this new approach offers as 
 well a new viewpoint on the old problem of mappings between smooth manifold
 s. This is joint work with FH Lin, A Segatti and C Wang.</p>
CONTACT:Johns Hopkins University
DTSTAMP:20260827T204607
DTSTART;TZID=America/New_York:20260414T134000
DTEND;TZID=America/New_York:20260414T144000
SEQUENCE:0
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