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SUMMARY:The Stable Bernstein theorem in R^5 - Paul Minter (Clay Institute\
 , Princeton\, Cambridge)
DTSTART:20240119T140000Z
DTEND:20240119T150000Z
UID:TALK210814@talks.cam.ac.uk
CONTACT:Dr Greg Taujanskas
DESCRIPTION:Short Version: I will explain why immersed stable minimal hype
 rsurfaces in R5 are flat. This is joint work with Otis Chodosh\, Chao Li\,
  and Douglas Stryker.\n\nLong Version: The stable Bernstein problem asks w
 hether an immersed stable minimal hypersurface in Rn is necessarily flat. 
 If true the result implies\, for example\, a priori curvature estimates fo
 r immersed stable minimal hypersurfaces in Riemannian n-manifolds.\n\nAn i
 mportant special case of the result\, known as the Bernstein problem\, ask
 s the same question except for minimal graphs over a hyperplane (such grap
 hs are in fact locally area minimising). The Bernstein problem was resolve
 d in full in the 1960's by works of Fleming\, De Giorgi\, Almgren\, Simons
 \, Bombieri\, and Giusti\, in turn driving a lot of progress in the develo
 pment of geometric measure theory. It was shown that such a minimal graph 
 must be flat if it lies in R8 or lower\, whilst counterexamples exist in R
 9 and higher. The stable Bernstein theorem has counterexamples in R8 and a
 bove for related reasons.\n\nThe stable Bernstein theorem in its full gene
 rality remained essentially open until recently. The works of Schoen—Sim
 on—Yau (1975) and Bellettini (2023) establish the result up to R7 assumi
 ng the minimal hypersurface has Euclidean volume growth. The full R3 case 
 was resolved independently by works of Fischer-Colbrie—Schoen\, do Carmo
 —Peng\, and Pogorelov\, all around 1980. In 2021\, Chodosh—Li proved t
 he stable Bernstein theorem in R4\, using techniques from non-negative sca
 lar curvature. They later found another proof using techniques from unifor
 mly positive scalar curvature utilising Gromov's mu-bubbles\, showing that
  the Euclidean volume growth assumption holds a priori.\n\nIn this talk\, 
 I will discuss recent work (joint with Otis Chodosh\, Chao Li\, and Dougla
 s Stryker) resolving the stable Bernstein problem in R5. Our proof uses in
 stead techniques from the study of uniformly positive bi-Ricci curvature. 
 We show that a suitable conformal change of metric (the Gulliver—Lawson 
 metric) has uniformly positive bi-Ricci curvature in a spectral sense. Thi
 s allows us to construct (warped) mu-bubbles with uniformly positive Ricci
  curvature in a spectral sense\, from which we can establish a Bishop volu
 me-comparison theorem (which is a subtle adaptation of a technique from Br
 ay\; indeed\, our proof seems to break down in any higher dimensional situ
 ation). These ingredients allow us to establish a priori Euclidean volume 
 growth for the minimal hypersurface allowing us to prove the result.
LOCATION:MR15
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