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BEGIN:VEVENT
UID:ca7efe572e0c0f33cf2fd24c368ab633
CATEGORIES:Geometric Analysis Seminar
CREATED:20241027T203144
SUMMARY:Weak Inverse Mean Curvature Flow in Hyperbolic Space
LOCATION:Hill-705
DESCRIPTION:Abstract: Inverse mean curvature flow (IMCF) is a geometric flow that expan
 ds hypersurfaces by mean curvature. IMCF has many geometric applications, b
 ut a key obstacle to these is the formation of finite-time singularities. T
 o deal with this, Huisken and Ilmanen developed a theory of weak solutions 
 of IMCF which flow beyond these singularities and out to infinity. The flow
  surfaces of weak IMCF may not be smooth and may vary discontinuously in th
 e time variable, a phenomenon known as a "jump". Furthermore, the asymptoti
 c behavior of weak solutions is poorly understood in manifolds with negativ
 e curvature, e.g. hyperbolic space. In this talk, I will show that a weak I
 MCF in hyperbolic space eventually becomes a classical IMCF for arbitrary c
 losed initial data-- that is, the flow surfaces become smooth and the jumps
  cease after an explicit time depending on the initial hypersurface. The pr
 oof is based on an Alexandrov reflection method in the Poincare ball. Then,
  I will apply this regularity result to prove two Minkowski inequalities in
  hyperbolic space. One of these implies a Penrose-type inequality in genera
 l relativity.
X-ALT-DESC;FMTTYPE=text/html:<div data-olk-copy-source="MessageBody" style="border: 0px; font-style: nor
 mal; font-weight: 400; font-size: 15px; line-height: inherit; font-family: 
 'Segoe UI', 'Segoe UI Web (West European)', 'Segoe UI', -apple-system, Blin
 kMacSystemFont, Roboto, 'Helvetica Neue', sans-serif; margin: 0px; padding:
  0px; vertical-align: baseline; color: #424242; letter-spacing: normal; orp
 hans: 2; text-align: start; text-indent: 0px; text-transform: none; widows:
  2; word-spacing: 0px; white-space: normal; background-color: #ffffff;">Abs
 tract: Inverse mean curvature flow (IMCF) is a geometric flow that expands 
 hypersurfaces by mean curvature. IMCF has many geometric applications, but 
 a key obstacle to these is the formation of finite-time singularities. To d
 eal with this, Huisken and Ilmanen developed a theory of weak solutions of 
 IMCF which flow beyond these singularities and out to infinity. The flow su
 rfaces of weak&nbsp;IMCF may not be smooth and may vary discontinuously in 
 the time variable, a phenomenon known as a "jump". Furthermore, the asympto
 tic behavior of weak solutions is poorly&nbsp;understood in&nbsp;manifolds 
 with negative curvature,&nbsp;e.g. hyperbolic space.</div><div style="borde
 r: 0px; font-style: normal; font-weight: 400; font-size: 15px; line-height:
  inherit; font-family: 'Segoe UI', 'Segoe UI Web (West European)', 'Segoe U
 I', -apple-system, BlinkMacSystemFont, Roboto, 'Helvetica Neue', sans-serif
 ; margin: 0px; padding: 0px; vertical-align: baseline; color: #424242; lett
 er-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-t
 ransform: none; widows: 2; word-spacing: 0px; white-space: normal; backgrou
 nd-color: #ffffff;">&nbsp;</div><div style="border: 0px; font-style: normal
 ; font-weight: 400; font-size: 15px; line-height: inherit; font-family: 'Se
 goe UI', 'Segoe UI Web (West European)', 'Segoe UI', -apple-system, BlinkMa
 cSystemFont, Roboto, 'Helvetica Neue', sans-serif; margin: 0px; padding: 0p
 x; vertical-align: baseline; color: #424242; letter-spacing: normal; orphan
 s: 2; text-align: start; text-indent: 0px; text-transform: none; widows: 2;
  word-spacing: 0px; white-space: normal; background-color: #ffffff;">In thi
 s talk, I will show that a weak IMCF in hyperbolic space eventually becomes
  a classical IMCF for arbitrary closed initial data-- that is, the flow sur
 faces become smooth and the jumps cease after an explicit time depending on
  the initial hypersurface. The proof is based on an Alexandrov reflection m
 ethod in the Poincare ball. Then, I will apply this regularity result to pr
 ove two Minkowski inequalities in hyperbolic space. One of these implies a 
 Penrose-type inequality in general relativity.</div>
CONTACT:Brian D. Harvie
DTSTAMP:20260828T094306
DTSTART;TZID=America/New_York:20241105T145000
DTEND;TZID=America/New_York:20241105T155000
SEQUENCE:0
TRANSP:OPAQUE
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